Econometric Analysis of Production Networks with

Dominant Units�

M. Hashem PesaranDepartment of Economics & USC Dornsife INET, University of Southern California, USA

and Trinity College, Cambridge, UK

Cynthia Fan YangDepartment of Economics, University of Southern California

August 4, 2017

AbstractThis paper considers production and price networks with unobserved common

factors, and derives an exact expression for the rate at which aggregate �uctuationsvary with the dimension of the network. It introduces the notions of strongly andweakly dominant and non-dominant units, and shows that at most a �nite numberof units in the network can be strongly dominant. The pervasiveness of a networkis measured by the degree of dominance of the most pervasive unit in the network,and is shown to be equivalent to the inverse of the shape parameter of the power law�tted to the network outdegrees. New cross-section and panel extremum estimatorsfor the degree of dominance of individual units in the network are proposed andtheir asymptotic properties investigated. Using Monte Carlo techniques, the pro-posed estimator is shown to have satisfactory small sample properties. An empiricalapplication to US input-output tables spanning the period 1972 to 2007 is providedwhich suggests that no sector in the US economy is strongly dominant. The mostdominant sector turns out to be the wholesale trade with an estimated degree ofdominance ranging from 0.72 to 0.82 over the years 1972-2007.

Keywords: aggregate �uctuations, strongly and weakly dominant units, spatialmodels, outdegrees, degree of pervasiveness, power law, input-output tables, USeconomyJEL Classi�cations: C12, C13, C23, C67, E32

�We acknowledge helpful comments by Daron Acemoglu, Vasco Carvalho, Ron Smith, Martin Weidner,and participants at the Max King Conference at Monash University, the 2017 Asian Meeting of theEconometric Society, the third Vienna Workshop on High Dimensional Time Series in Macroeconomicsand Finance, the fourth annual conference of the International Association for Applied Econometrics(IAAE), and seminar participants at University of Southern California. We would also like to thankIda Johnsson for help with the compilation of input-output tables. Yang gratefully acknowledges theIAAE travel grant. Corresponding address: Department of Economics, University of Southern California,3620 South Vermont Avenue, Kaprielian Hall 300, Los Angeles, CA 90089, USA. E-mail addresses:pesaran@usc.edu (M. Hashem Pesaran), yang488@usc.edu (Cynthia Fan Yang).

1 Introduction

Over the past decade, there has been renewed interest in production networks and the rolethat individual production units (�rms/sectors) can play in propagation of shocks acrossthe economy. This literature builds on the multisectoral model of real business cyclespioneered by Long and Plosser (1983), and draws from a variety of studies on social andeconomic networks, including network games, cascades, and micro foundations of macrovolatility. Notable theoretical contributions in this area include Acemoglu, Carvalho,Ozdaglar, and Tahbaz-Salehi (2012), Horvath (1998, 2000), Gabaix (2011), Acemoglu,Ozdaglar, and Tahbaz-Salehi (2016), and Siavash (2016). Empirical evidence for suchpropagation mechanism is presented in Foerster, Sarte, and Watson (2011), Acemoglu,Akcigit, and Kerr (2016), and Carvalho, Nirei, Saito, and Tahbaz-Salehi (2016). One im-portant issue in this literature relates to conditions under which sector-speci�c shocks arelikely to have lasting aggregate (macro) e¤ects. Similar issues arise in �nancial networkswhere it is of interest to ascertain if an individual bank can be considered as �too bigto fail�. Recent reviews are provided by Carvalho (2014) and Acemoglu, Ozdaglar andTahbaz-Salehi (2016).In this paper we consider a production network with unobserved common technological

factors, and derive an associated price network which is dual to the production network,which we use to derive an exact characterization of the e¤ect of sector-speci�c shocks onaggregate output. We show that sector-speci�c shocks have aggregate e¤ects if there are�dominant�sectors in the sense that their outdegrees are not bounded in the number ofproduction units, N , in the economy. The outdegree of a sector is de�ned as the shareof that sector�s output used as intermediate inputs by all other sectors in the economy.The degree of dominance (or pervasiveness) of a sector is measured by the exponent �that controls the rate at which the outdegree of the sector in question rises with N .This measure turns out to be the same as the exponent of cross-sectional dependenceintroduced in Bailey et al. (2016), for the analysis of cross-section dependence in paneldata models with large cross-section and time dimensions.Our approach di¤ers from Acemoglu et al. (2012) in three important respects. First,

we provide a more general setting that allows for unobserved common factors and derivea spatial model in sectoral prices that can be taken directly to the data. We establish aone-to-one relationship between the pervasiveness of price shocks and aggregate outputshocks. Second, Acemoglu et al. (2012) express the aggregate output as a reduced formfunction of the sector-speci�c shocks, based on which they are only able to derive alower bound to the decay rate of sector-speci�c shocks on aggregate outcomes. Theyconsider the �rst- and second-order e¤ects, and acknowledge that ignoring higher-orderinterconnections might bias the results. In contrast, the present paper provides an exactexpression for the e¤ects of sector-speci�c shocks on aggregate �uctuations, and showsthat its rate of decay only depends on the extent to which the dominant unit (sector) ispervasive, namely the one with the largest �, denoted by �max. We derive upper as well aslower bounds for the rate of convergence of the variability of aggregate output in termsof N , and show that these bounds converge at the same rate, and thus establish an exactrate of convergence for aggregate output variability. Finally, Acemoglu et al. (2012) do notidentify the dominant unit(s). Instead, they approximate the tail distribution, for some

1

given cut-o¤ value, of the outdegrees by a power law distribution and provide estimatesfor the shape parameters. By contrast, we propose a nonparametric approach, whichis applicable irrespective of whether the outdegrees are Pareto distributed, and does notrequire knowing the cut-o¤value above which the Pareto tail behavior begins. The inverseof the proposed estimator of �max is an extremum estimator of the shape parameter ofthe Pareto distribution, �. It is simple to compute and is given by the average log of thelargest outdegree relative to all other outdegrees, scaled by the size of the network, N .In addition, our approach also allows us to identify the most dominant units and theirdegrees of dominance, �(i), where �(1) = �max � �(2) � :::, for all �(i) > 1=2.Small sample properties of the extremum estimator are investigated by Monte Carlo

techniques and are shown to be satisfactory. A comparison of the estimates of the shapeparameter � based on Pareto distribution with the estimates based on the inverse of theextremum estimator of �max, shows that the latter performs much better, particularlywhen N is large (300+). Furthermore, the extremum estimator is shown to perform welleven under a Pareto tail distribution, whereas the commonly used estimators of the shapeparameter, �, display substantial biases if the true underlying distribution is non�Pareto.Application of our estimation procedure to US input-output tables over the period

1972-2002 yields yearly estimates of �max that lie between 0:72 and 0:82. These estimatesare by and large close to the inverse of the estimates of the shape parameter � considered inAcemoglu et al. (2012) when a 20% cut-o¤ value is used, although the log-log regressionestimates of � tend to be highly sensitive to the choice of the cut-o¤ values and thedi¤erent orders of interconnections considered. To provide more reliable estimates, wealso conduct panel estimation and �nd that the largest estimate of �max is about 0:76 forthe sub-sample covering 1972-1992 and 0:72 for the sub-sample covering 1997-2007. Quiteremarkably, we �nd that estimates of �max and the identity of the dominant sector arerather stable throughout the period from 1972 to 2007, with the wholesale trade sectoridenti�ed as the most dominant sector for all years except for the year 2002 when thewholesale trade is estimated to be the second most dominant sector. Our estimates alsosuggest that no sector in the US economy is strongly dominant, which requires the valueof �max to be close to unity, whilst the largest estimate we obtain is around 0:8. Overall,our analyses support the view that sector-speci�c shocks have some macro e¤ects, but wedo not �nd such e¤ects to be su¢ ciently strong.The rest of the paper is organized as follows. Section 2 presents the production net-

work. Section 3 derives the associated price network. Section 4 introduces the concepts ofstrongly and weakly dominant, and non-dominant units, and network pervasiveness. Sec-tion 5 derives exact conditions under which micro (sectoral) shocks can lead to aggregate�uctuations. Section 6 shows the relation between the degree of network pervasivenessand the shape parameter of the power law distribution. Section 7 introduces the extremumestimator, derives its asymptotic distribution, and shows its robustness to the choice ofthe underlying distribution. Section 8 provides evidence on the small sample propertiesof the alternative estimators of �max using a number of Monte Carlo experiments. Section9 presents the empirical application, and Section 10 concludes. Some of the mathematicaldetails and a brief discussion of data sources are provided in the Appendix. An extensionof the extremum estimator to large N and T panels, and additional Monte Carlo resultsare provided in an Online Supplement.

2

Notations: The total number of cross section units (sectors) in the economy is de-noted by N , which is then decomposed into m dominant units and n non-dominantunits. The number of dominant units is also decomposed into strongly dominant unitsand weakly dominant units. (See De�nition 1). If ffNg1N=1 is any real sequence andfgNg1N=1 is a sequence of positive real numbers, then fN = O(gN) if there exists a pos-itive �nite constant K such that jfN j =gN � K for all N . fN = o(gN) if fN=gN ! 0;as N ! 1. If ffNg1N=1 and fgNg

1N=1 are both positive sequences of real numbers, then

fN = (gN) if there exists N0 � 1 and positive �nite constants K0 and K1, such thatinfN�N0 (fN=gN) � K0; and supN�N0 (fN=gN) � K1. %(A) is the spectral radius of theN � N matrix A = (aij), de�ned as %(A) = max fj�ij ; i = 1; 2; :::; Ng, where �i is aneigenvalue of A and j�1(A)j � j�2(A)j � � � � � j�N(A)j. kAk1 = max

1�i�N

PNj=1 jaijj

and kAk1 = max1�j�N

PNi=1 jaijj are the maximum row sum norm and the maximum column

sum norm of matrix A, respectively. K is used for a generic �nite positive constant notdepending on N . �i denotes the degree of dominance (or pervasiveness) of unit i in anetwork, where i = 1; 2; :::; N; and 0 � �i � 1.

2 Production network

To show how the two strands of literature on production networks and cross-sectionaldependence are related, we begin with a panel version of the input-output model developedin Acemoglu et al. (2012). Our goal is to provide an exact characterization of the e¤ectof unit-speci�c shocks on aggregate output. We assume that production of sector i attime t, qit, is determined by the following Cobb-Douglas production function subject toconstant returns to scale

qit = e�uitl�it

NYj=1

q�wijij;t ; for i = 1; 2; : : : ; N ; t = 1; 2; :::; T; (1)

where the productivity shock, uit, is given by

uit = "it + ift;

and is composed of a sector-speci�c shock, "it, and a common technological factor, ft. Thefactor loading, i, measures the importance of technological change on sector i. FollowingBailey et al. (2016), we denote the cross-section exponent of the factor loadings by � ,de�ned by the value of � that ensures

limN!1

N�� NXi=1

j ij = c > 0; (2)

where c is a �nite constant, bounded in N . The standard factor model sets � = 1, andtreats the common factor as �strong�or �pervasive�, in the sense that changes in ft a¤ectsall sectors of the economy. But in what follows we shall also consider cases where � < 1.In the case where the factor loadings are random we shall assume that E( i) = 6= 0,

3

and V ar( i) = �2 > 0: The analysis can be easily extended to allow for multiple factorswithout additional complexity.In line with Acemoglu et al. (2012), we shall assume that the sector-speci�c shocks

are cross-sectionally independent with zero means and �nite variances, V ar("it) = �2i ,such that 0 < �2 < �2i < ��2 < K < 1. The independence assumption is not necessaryand can be relaxed by assuming that the errors are cross-sectionally weakly dependent.We also assume that "it are serially uncorrelated, although this is not essential for ourmain theoretical results, and without loss of generality we assume that common andsector-speci�c shocks are uncorrelated.Returning to the other factors in the production function, lit denotes the labor input,

qij;t is the amount of output of sector j used in production of sector i, � denotes the shareof labor, � = 1� �, and �wij is the share of jth output in the ith sector. The amount of�nal goods, cit, are de�ned by

cit = qit �NXj=1

qji;t; i = 1; 2; : : : ; N; (3)

which are consumed by a representative household with the Cobb-Douglas preferences

u (c1t;c2t; : : : ; cNt) = ANYi=1

c1=Nit ; A > 0: (4)

We further assume that the aggregate labor supply, lt, is given exogenously and labormarkets clear

lt =NXi=1

lit: (5)

Let P1t; P2t; : : : ; PNt be the sectoral equilibrium prices, Waget the equilibrium wagerate, and denote their logarithms by pit = log (Pit), !t = log (Waget). Then it can beshown that in the competitive equilibrium the logarithm of real wage, which is taken asa measure of GDP in the literature, is given by

!t � pt = �+ (�0N ) ft + �0N"t; (6)

where pt is the aggregate log price index de�ned by,

pt = N�1NXi=1

pit = N�1� 0Npt; (7)

pt = (p1t; p2t; : : : ; pNt)0, = ( 1; 2; : : : ; N)

0, "t = ("1t; "2t; : : : ; "Nt)0, and

�N = (�1; �2; : : : ; �N)0 =

�

N(IN � �W0)

�1�N ; (8)

where W is the N � N matrix W = (wij) ; and �N is an N � 1 vector of ones. � is a

4

constant independent of ft and "t; which is given by

� = ��1

"� log (�) + � log (�) + �

NXi=1

NXj=1

�iwij log (wij)

#:

The (log) real-wage equation, (6), generalizes equation (3) in Acemoglu et al. (2012)by allowing for time variations in prices and technologies. By normalizing pt such thatpt = �� and ignoring the common factor, Acemoglu et al. (2012) concentrate on !t =�0N"t; as a measure of aggregate output, and refer to �N as the �in�uence vector�(theirequation (4)), and show that �i � 0, and � 0N�N = 1.1 They measure aggregate volatilityby the standard deviation of aggregate output, namely [V ar (�0N"t)]

1=2 ; and focus on theasymptotic properties of �0N"t, as N !1. Since V ar (�0N"t) = �0NV ar ("t)�N , it followsthat

��2 (�0N�N) � V ar (�0N"t) � �2 (�0N�N) ;

and hence the asymptotic properties of V ar (�0N"t) is governed by that of �0N�N . Ace-

moglu et al. (2012, p.2009) derive a lower bound for �0N�N and show that2

�0N�N � k�;0N�1 + k�;1N

�2NXj=1

d2j ; (9)

where k�;0 and k�;1 are �nite constants that depend on �, and dj is de�ned by dj =PNi=1wij; which is the j

th column sum ofW. Therefore, to analyze the limiting behaviorof �0N�N , it is su¢ cient to consider the limiting behavior of N

�2PNj=1 d

2j . This analysis

is carried out in some detail by Acemoglu et al. (2012). But as we shall see, it is alsoimportant to consider the limiting behavior of individual column sums of W, and inparticular to identify the ones that rise with N , as distinguished from those that arebounded in N .

3 Price network

Instead of analyzing the aggregate output directly in terms of the sector-speci�c shocks,we derive a price network which is dual to the production network discussed in Section2. By doing so, we are able to obtain an exact expression for the decay rate of aggregatevolatility, rather than just a lower bound. Given sector prices, P1t; P2t; : : : ; PNt, and thewage rate, Waget, solving sector i�s problem leads to

qij;t =�wijPitqit

Pjt; (10)

and

lit =�PitqitWaget

: (11)

1See Appendix A of Acemoglu et al. (2012).2These authors also consider higher-order interconnection terms which they include on the right-hand-

side of �0N�N , but these terms are dominated by N�2PN

j=1 d2j .

5

Substituting the above results in (1) and simplifying yields

pit = �

NXj=1

wijpjt + �!t � bi � � ( ift + "it) ; (12)

where the price-speci�c intercepts, bi, depend only on �, � and the weight matrixW,

bi = � log (�) + � log (�) + �

NXj=1

wij log(wij); (13)

for i = 1; 2; : : : ; N . In cases where wij = 0, we set wij log (wij) = 0 as well. In matrixnotation the �price network�, (12), can be written as

pt = �Wpt + �!t�N � (b+ � ft + �"t) ; (14)

where b = (b1; b2; : : : ; bN)0 :

A dual to the price equation in (12) can also be obtained using (10) in (3) to obtain

Sit = �NXj=1

wjiSjt + Cit; (15)

where Cit = Pitcit, and Sit = Pitqit is the sales of sector i. The sales equation, (15), canalso be written as

St = �W0St +Ct; (16)

where St = (S1t; S2t; : : : ; SNt)0 and Ct = (C1t; C2t; : : : ; CNt)

0. Note that W enters as itstranspose,W0, in (16) as compared to the price equations in (14).Aggregating (11) over i, we have

Waget

NXi=1

lit = �NXi=1

Pitqit;

or

ltWaget = �NXi=1

Sit = �� 0NSt: (17)

Also using (16)St = (IN � �W0)

�1Ct; (18)

where (IN � �W0)�1 is known as the Leontief inverse.3 Using (18) in (17) now yields thefollowing expression for the total wage bill,

ltWaget = �� 0N (IN � �W0)�1Ct: (19)

3A proof that the Leontief matrix is invertible even in the presence of dominant units is provided inLemma 1 of Appendix A.

6

Similarly, solving (14) for the log-price vector, pt, and applying Lemma 1 in AppendixA we have

pt = �!t (IN � �W)�1 �N � � (IN � �W)�1 �t, (20)

where �t = ��1b + ft + "t. Then the aggregate log price index, pt, de�ned in (7), isgiven by

pt =h �N� 0N (IN � �W)�1 �N

i!t �

�

N� 0N (IN � �W)�1 �t: (21)

But since wij � 0, W�N = �N , and 0 < � < 1, then (IN � �W)�1 �N = �N=�, andhence (21) can also be written as

!t � pt = �0N�t; (22)

where �N is in�uence vector given by (8). Now let

xt = pt � !t�N ; (23)

and rewrite (14) in terms of log price-wage ratios, xt, as

xt = �Wxt � b� � ( ft + "t) : (24)

Equation (24) represents a �rst-order spatial autoregressive (SAR(1)) model with anunobserved common factor.Consider now the following simple average over the units, xit, for i = 1; 2; :::; N , in the

above networkxN;t =

1

N� 0Nxt = � (!t � �pt) ;

which is the negative of the aggregate output measure, de�ned by (6). Also, using (22)we have

!t � �pt = �xN;t = ��1 (�0Nb) + (�0N ) ft + �

0N"t; (25)

which fully speci�es the dependence of aggregate output on both common and sector-speci�c shocks.Note that equations (19) and (25) are dual of each other. (19) gives the total wage

bill in terms of a weighted sum of consumption expenditures, with the weights givenby � (IN � �W)�1 �N . Whilst (25) gives the log of the real wage rate in terms of thecommon and aggregate sectoral shocks, namely ft and �0N"t. Recall that common andsectoral shocks are assumed to be uncorrelated. The key issue is how much of the cyclical�uctuations in (log) real wages, V ar (xN;t) ; is due to common shocks (�0N )

2 V ar (ft) ;and how much is due to sectoral shocks, V ar (�0N"t).There are two advantages in directly focusing on the price network model, (24). First,

it allows us to relate the production network to the literature on spatial econometricsthat should facilitate the econometric analysis of production networks, and allows us toaddress more easily the issues of identi�cation and estimation of the structural parameters�, and �2i , for i = 1; 2; :::; N .

4 The direct use of the SAR model, (24), also enables us

4For example, see the recent contributions of Bai and Li (2013) and Yang (2016) on estimation ofSAR models with unobserved common factors.

7

to provide exact bounds on V ar (xN;t) = V ar (!t � �pt) rather than the lower boundsobtained by Acemoglu et al. (2012). Instead, by considering the price network explicitlywe are able to show that at most only a few sectors can have signi�cant aggregate e¤ects,and these sectors are those with outdegrees that rise with N . The rate at which theoutdegrees rise with N could very well di¤er across sectors and it is important that suchsectors are identi�ed and their empirical contribution to aggregate �uctuations evaluated.

4 Degrees of dominance of units in a network andnetwork pervasiveness

Consider a network represented by the N � N adjacency matrix W = (wij), wherewij � 0 for all i and j, and W is row-normalized such that

PNj=1wij = 1, for all i.

Denote the jth column ofW by w�j and the associated column sum by dj = � 0Nw�j, theoutdegree of unit j. The outdegree is one of many network centrality measures consideredin the literature. The most widely used centrality measures is degree centrality, whichrefers to the number of ties a node has, and in a directed network can be classi�ed intoindegree and outdegree. The indegree counts the number of ties a node receives, andthe outdegree counts the number of ties a node directs to others. In this paper, we arefocusing on how the weighted outdegree vary with N and normalize the weighted indegree(row sums of W) to one, because we are interested in studying the in�uence of a unitto other downstream units. Other centrality measures, including closeness, betweenness,and eigenvector centralities, are not relevant for our purpose, since we aim to characterizethe e¤ects of idiosyncratic shocks to a unit on some aggregate measure of the network,rather than the pattern of interdependencies of the network. To this end, we introducethe notions of strongly and weakly dominant units in the following de�nition. We considerunits with nonzero outdegrees and assume throughout that dj > 0, for all j.

De�nition 1. (�-dominance) We shall refer to unit j of the row-standardized networkW = (wij � 0) as �j-dominant if its (weighted) outdegree, dj =

PNi=1wij > 0, is of order

N �j , where �j is a �xed constant in the range 0 � �j � 1. More speci�cally,

dj = �jN�j ; for j = 1; 2; :::; N; (26)

where �j is a �xed positive constant which does not depend on N . The unit j is saidto be strongly dominant if �j = 1, weakly dominant if 0 < �j < 1, and non-dominant if�j = 0. We refer to �j as the degree of dominance of unit j in the network.

Remark 1. Since dj > 0 for all j, we must have �j > 0 and �min = min (�1; �2; :::; �N) >0. It is also worth noting that �j is identi�ed by requiring that �j and �j do not dependon N .

In the standard case where the column sum ofW is bounded inN we must have �j = 0for all j, that is, all units are non-dominant. W will have an unbounded column sum if�j > 0 for at least one j. But due to the bounded nature of the rows ofW, not all columns

8

ofW can be �-dominant with �j > 0 for all j. To see this, let d = (d1; d2; :::; dN)0 =W0�N ,

and note that� 0Nd = �

0NW

0�N = N: (27)

Hence, there must exist 0 < �j < K <1 for j = 1; 2; :::; N; such that

NXj=1

�jN�j = N; (28)

for a �xed N and as N !1. Let �min = min (�1; �2; :::; �N), and note that

N =NXj=1

�jN�j � N�minN

�min ;

which in turn implies�minN

�min � 1: (29)

Since by assumption �min > 0 and �min � 0, it is clear that (29) cannot be satis�ed for allvalues of N unless �min = 0, which establishes that not all units in a given network canbe dominant. This result is summarized in the following proposition.

Proposition 1. Consider the network represented byW = (wij � 0), and assume thatWis row-standardized. Suppose that the outdegrees of the network, dj =

PNi=1wij, are non-

zero (dj > 0) and follow the power function, (26), with �j being the degree of dominanceof unit j in the network. Then not all units of the network can be �-dominant, with �j > 0for all j.

Let SN = N�1PNj=1 �jN

�j ; and note that since �j > 0 for all j, �min = minj(�j) > 0,and hence

SN = N�1NXj=1

�je�j lnN � �minN

�1NXj=1

e�j lnN . (30)

Now using a Taylor series expansion of e�j lnN ; we obtain

NXj=1

e�j lnN =NXj=1

"1 +

1Xs=1

�sj (lnN)s

s!

#

= N +

1Xs=1

(lnN)s

s!

NXj=1

�sj

!; (31)

which if substituted in (30) yields

SN � �min

241 + 1Xs=1

�PNj=1 �

sj

�(lnN)s

s!N

35 : (32)

9

Since SN = 1, and all the summands over s in (32) are nonnegative as �j � 0 and lnN > 0,it is necessary that�PN

j=1 �sj

�(lnN)s

s!N! 0; as N !1, for all s = 1; 2; 3; :::: (33)

Also note that for any �nite s, (lnN)s = (s!N)! 0; as N !1, and since1Xs=1

(lnN)s

s!N=N � 1N

! 1; as N !1;

then it must be that (lnN)s = (s!N) ! 0, as N ! 1, for all s, including s ! 1.Furthermore, since 0 � �j � 1 then

NXj=1

�sj �NXj=1

�j; for s � 1; (34)

and �PNj=1 �

sj

�(lnN)s

s!N�

NXj=1

�j

!(lnN)s

s!N:

Hence, for conditions in (33) to be met it is su¢ cient that f�jg is summable, namely

NXj=1

�j < K <1: (35)

As we shall see, this condition plays an important role in the proof of consistency of theextremum estimator proposed in the sub-section 7.2 below.Suppose now that m units are strongly dominant with �j = 1, and the rest are non-

dominant with �j = 0. Then using (32) we have

SN � �min

"1 +m

1Xs=1

(lnN)s

s!N

#= �min

�1 +m

�N � 1N

��;

and since SN = 1, it follows that m cannot rise with N , and must be a �xed integer.In the case where m units are dominant with �j > 0, then m must be �nite if the

summability condition given by (35) is to hold. For example, suppose that only m units

are dominant. ThenNPj=1

�j � m�min > 0, and from the summability condition (35) we

have K >NPj=1

�j � m�min, from which it follows that m � K=�min which is bounded in N .

These �ndings are summarized in the next proposition.

Proposition 2. Consider the network represented by W = (wij � 0), and assume thatW is row-standardized, and the outdegrees of the network, dj =

PNi=1wij, are non-zero

10

(dj > 0). Then the number of strongly dominant units must be �xed and cannot rise withN . Moreover, if f�jg are summable,

PNj=1 �j < K < 1, then the number of dominant

units with �j 6= 0 must be �nite, where �j is the degree of dominance of unit j in thenetwork.

Remark 2. Analogous results have also been found in Chudik, Pesaran, and Tosetti(2011) regarding the possible number of strong factors, and in Chudik and Pesaran (2013)on the number of dominant units in large dimensional vector autoregressions.

Using the concept of �-dominance of units in a given network, we now introduce theidea of network pervasiveness which is relevant for characterization of the degree to whichshocks to an individual unit di¤use across the other units in the network.

De�nition 2. (Network pervasiveness) Degree of pervasiveness of a given row-standardizednetwork, W = (wij � 0,

PNj=1wij = 1), is de�ned by �max = max (�1; �2; :::; �N) ; where

�j is the degree of dominance of its jth unit.

Using the above concepts we now consider the rate at which V ar (xN;t) varies withN , and show that it is governed by the pervasiveness of the network, measured by �max,and the exponent of the common factor, � ; de�ned by (2). For unit-speci�c shocks todominate the macro or common factor shocks we need �max > � > 1=2. We also showhow our measure of network pervasiveness, �max, is related to �; the shape parameter ofthe Pareto distribution �tted to the ordered outdegrees, d(i), for i = 1; 2; :::; N .

5 Price networks with one dominant unit and aggre-gate �uctuations

Consider the price network (24), and assume that it contains 1 dominant unit and n =N � 1 non-dominant units. The analysis can be readily extended to networks with mdominant units (m �xed), but to simplify the exposition here we con�ne our analysis tonetworks with one dominant unit. (The derivations for the general case is provided inAppendix B). Without loss of generality suppose the �rst element of xt, namely x1t, isthe dominant unit, and write (24) in the partitioned form as (setting w11 = 0)�

x1tx2t

�=

�0 �w0

12

�w21 �W22

��x1tx2t

�+

�g1tg2t

�; (36)

where x2t = (x2t; x3t; :::; xNt)0, w21 = (w21; w31; :::; wN1)0, w12 = (w12; w13; :::; w1N)0, g2t =(g2t; g3t; :::; gNt)

0, and git = �bi��( ift+ "it); for i = 1; 2; :::; N . W22 is the n� n weightmatrix associated with the n non-dominant units and is assumed to satisfy the conditionj�j kW22k1 < 1. Furthermore, note that since

W�N =

�0 w0

12

w21 W22

��1� n

�=

�1� n

�;

then w012� n = 1, and � n � w21 =W22� n. The latter result states that the ith row sum

of W22 is given by 1 � wi1 � 1, and considering that 0 � wi1 < 1, then we must have

11

kW22k1 � 1, which also establishes that %(W22) � 1, where %(A) denotes the spectralradius of A. Under the assumption that j�j < 1, by Lemma 1 in Appendix A the systemof equations (36) has a unique solution given by�

x1tx2t

�=

�1 ��w0

12

��w21 S22

��1�g1tg2t

�(37)

= S�1(�)gt;

where S22 = In � �W22. In addition, since j�j kW22k1 < 1, it follows from Lemma 2 inAppendix A that S�122 has bounded row and column norms.For future reference also note that the (1; 1)th element of S�1(�) is given by ��11 , where

�1 = 1� �2w012S

�122w21 6= 0: (38)

Finally, to allow unit 1 to be �-dominant we consider the following exponent formulation

d1 =NXi=2

wi1 = �1N�1 ; (39)

where d1 is allowed to rise with N , with �1 > 0 and 0 < �1 � 1. Recall that �1 and �1can not vary with N .5

The system of equations (36) can now be solved for x2t in terms of x1t, namely (recallthat by assumption j�j kW22k1 < 1)

x2t = x1t��S�122w21

�+ S�122 g2t; (40)

and6

x1t = ��11�g1t + �w0

12S�122 g2t

�: (41)

Using the above in (40), we now have

x2t = (�=�1)�g1t + �w0

12S�122 g2t

�S�122w21 + S

�122 g2t: (42)

The �rst term of x2t refers to the direct and indirect e¤ects of the dominant unit, and thesecond term relates to the network dependence of the non-dominant units.Our primary focus is the extent to which shocks to individual units a¤ect aggregate

measures over the network. A standard aggregate measure is cross-section averages of xitover i = 1; 2; :::; N . Here we consider the simple average

xN;t =x1t +

PNi=2 xit

N=x1t + �

0nx2t

N;

but our analysis equally applies to weighted averages, x�N;t =PN

i=1$ixit, so long as theweights $i are granular in the sense that $i = O (N�1). Using (40) and (41) we have

xN;t =x1t + �

0nS

�122 (�w21x1t � b2 � �"2t � � 2ft)

N;

5The exponent formulation of column sums in (39) will be compared and contrasted to the power lawspeci�cation favored in the literature in Section 7 below.

6In deriving (41), it is required that �1 6= 0. This condition is met since the N � N matrix on theright-hand-side of (37) is non-singular.

12

where b2 = (b2; b3; :::; bN)0 and 2 = ( 2; 3; :::; N)0. Hence

xN;t = N�1 (�an + �nx1t � � nft � ��0n"2t) ; (43)

where an = � 0nS�122 b2; �

0n = �

0nS

�122 , and

�n = 1 + ��0nw21; (44)

n = �0n 2: (45)

The �rst term of (43), N�1an, is bounded in N , since kW22k1 � 1 and � kW22k1 < 1,and as a result S�122 will have bounded row and column norms by Lemma 2 in AppendixA. The second term captures the e¤ect of the dominant unit. The third term is dueto the common factor, ft, and the �nal term represents the average e¤ects of the microproductivity shocks. N�1�n is the in�uence vector associated with the non-dominantunits. It is analogous to the in�uence vector de�ned by (8) which applies to all units.Starting with the �nal term of (43), we �rst note that

�2N�2�0n�n � V ar�N�1�0n"2t

�� ��2N�2�0n�n; (46)

where �0n = (�2; �3; :::; �N) is an n � 1 vector of column sums of S�122 and has boundedelements. Furthermore, since

�0n = �0n + �� 0nW22 + �2� 0nW

222 + :::;

� > 0 and wij � 0, then �min = min(�2; �3; :::; �N) > 1, and �max = max(�2; �3; :::; �N) <K <1: Hence,

1 < �2min � N�1�0n�n � �2max < K <1;

and N�1�0n�n is bounded from below and above by �nite non-zero constants. Using thisresult in (46) we also have

�2 < NV ar�N�1�0n"2t

�< ��2�2max <1;

which establishes thatV ar

�N�1�0n"2t

�=

�N�1� ; (47)

where (N�1) denotes the convergence rate of V ar (N�1�0n"2t) in terms of N , and shouldbe distinguished from the O (N�1) notation, which provides only an upper bound onV ar (N�1�0n"2t).Next, using (41) we have

Cov�x1t; N

�1�0n"2t�= �����11 N�1w0

12H22� n; (48)

andCov (x1t; ft) = ��

���11 1 + ���11 h2

�V ar (ft) ; (49)

where H22 = S�122V22;"S0�122 , V22;" = diag (�22; �

23; :::; �

2N), and h2 = w0

12S�122 2: It then

follows that overall (recalling that ft and "it are independently distributed), we have

V ar (xN;t) = N�2�2nV ar (x1t)� 2�N�2�nCov (x1t;�0n"2t) + �2N�2V ar (�0n"2t)

+�2N�2�nV ar (ft) ; (50)

13

where�n = 2n + 2 n�n�

�11 1 + 2� n�n�

�11 h2:

Also, using (41) we have

V ar (x1t) = ��21 �2�� 21 + �2h22

�V ar(ft) + �21

�+ ��21 �2�2w0

12H22w21; (51)

which is easily seen to be bounded in N .A number of results can now be obtained from (50). First, without a common factor

and a dominant unit, V ar (xN;t) = (N�1), and the e¤ects of idiosyncratic shocks on xN;twill vanish at the rate of N�1=2; as N !1. This rate matches the decay rate of shocks inmodels without a network structure, namely even if we setW = 0. Therefore, for microshocks to have macroeconomic implications there must be at least one dominant unit inthe network. To see this consider now the case where there is no common factor but thenetwork includes a dominant unit. Then using (47) and (50) we have

V ar (xN;t) = N�2�2nV ar (x1t)� 2�N�2�nCov (x1t;�0n"2t) +O

�N�1� : (52)

Recall that V ar(x1t) is bounded in N , and consider the limiting properties of N�1�nde�ned by (44). Since

N�1 + �min�N�1d1 � N�1�n � N�1 + �max�N

�1d1; (53)

where 1 � �min � �max < K, then the asymptotic behavior of N�1�n depends on theway the outdegree of the dominant unit, namely d1, varies with N . Using the exponentspeci�cation given by (26), d1 = �1N

�1, it follows that

N�1 + �min��1N�1�1 � N�1�n � N�1 + �max��1N

�1�1; (54)

which leads toN�1�n = (N �1�1); 0 < �1 � 1: (55)

Consider now the second term of (52), and note from (48) that

jCov (x1t;v0n"2t)j �����(1� �)�

�1

����N�1 kw012k1

S�122 1 kV22;"k1 k�nk1 = O�N�1� ;

since kw012k1 = kw12k1 =

PNi=2w1i = 1,

S�122 1 < K, kV22;"k1 = ��2 < K, andk�nk1 = �max < K: Using the above results in (52) we have

V ar (xN;t) = (N2�1�2) +O(N �1�2) +O�N�1� ;

which simpli�es to (since �1 � 1)

V ar (xN;t) = (N2�1�2) +O�N�1� ; (56)

and henceV ar (xN;t) = (N2�1�2); if �1 > 1=2: (57)

14

This is the main result for the analysis of macro economic implications of micro shocks,and is more general than the one established by Acemoglu et al. (2012) who only providea lower bound on the rate at which aggregate volatility changes with N .It is also instructive to relate N�1�n to the �rst- and higher-order network connections

discussed in Acemoglu et al. (2012). Expanding the terms of the inverse S�122 , N�1�n can

also be written as

N�1�n = N�1 �1 + �� 0nw21 + �2� 0nW22w21 + �3� 0nW222w21 + :::

�;

where N�1�� 0nw21 = �N�1d1 represents the e¤ects of the �rst-order network connectionson �N , N�1�2� 0nW22w21, the e¤ects of the second-order network connections and so on.But in view of (53) and (55) all these higher order interconnections (individually andtogether) at most behave as (N �1�1).Therefore, the rate at which micro shocks in�uence the macro economy depends on

�1, which measures the strength of the dominant unit. But it should be noted from (56)that to ensure a non-vanishing variance, V ar (xN;t) > 0; as N ! 1, we need a value of�1 = 1. When 1=2 < �1 < 1, the network accentuates the di¤usion of the idiosyncraticshocks across the network but does not lead to lasting impacts. No network e¤ects ofunit-speci�c shocks can be identi�ed when �1 � 1=2. Hence, for the dominant unit tohave any impact over and above the standard rates of diversi�cation of micro shocks onxN;t, we need �1 > 1=2.

Remark 3. The �nding that �1 cannot be distinguished from zero if �1 < 1=2 is alsorelated to the study by Bailey et al. (2016), who show that the exponent of cross-sectionaldependence, �, can only be identi�ed and consistently estimated for values of � > 1=2.

Suppose now that the network is subject to common shocks without a dominant unit.In this case we have

V ar (xN;t) = �2N�2 2nV ar (ft) +�N�1� ;

and the rate of convergence of xN;t is determined by the strength of the factor as givenby N�2 2n. Using (45) we have (recall that % (W22) � 1);

N�1 n = N�1� 0nS�122 2

= N�1 �� 0n 2 + �� 0nW22 2 + �2� 0nW222 2 + :::

�:

By a similar line of reasoning as before, it is then easily seen that N�1 n = �N � �1

�,

where � (already de�ned by (2)) is the cross-section exponent of the factor loadings, i, and measures the degree to which the common factor is pervasive in its e¤ects onsector-speci�c productivity.Finally, suppose that the production network is subject to a common factor as well as

containing a dominant unit. Then for �1 > 1=2 and � > 1=2 we have7

V ar (xN;t) = �N2�1�2

�+

�N2� �2

�+

�N�1� : (58)

7Note that for values of �1 and � < 1=2 the third term in (58) will dominate the network and thecommon factor e¤ects.

15

It is clear that the relative importance of the dominant unit and the common factordepends on the relative magnitudes of �1 and � . We need estimates of these exponentsfor a further understanding of the relative importance of macro and micro shocks inbusiness cycle analysis.Allowing for multiple factors and multiple dominant units does not alter the main

results, and the general expression in (58) will continue to apply, with the di¤erence that� and �1 in such a general setting will refer to the maximum of the exponents of thefactors and �max = max(�1; �2; :::; �N).

6 �max and � measures of network pervasiveness

The degree of network pervasiveness, �max, de�ned in De�nition 2, is related to �, theshape parameter of the power law assumed by Acemoglu et al. (2012, De�nition 2) forthe outdegree sequence, fd1; d2; :::; dNg. To see this we �rst use the speci�cation of theoutdegrees given by (26) in (9) to obtain

�0N�N � k�;0N�1 + k�;1N

�2mXj=1

�2jN2�j + k�;1

N �m

N2

NX

j=m+1

�2j= (N �m)

!;

where (N �m)�1PN

j=m+1 �2j = O(1). Also, recall that m must be �nite if f�jg is to be

summable (Proposition 2), and therefore

N�2mXj=1

�2jN2�j � m�2maxN

2(�max�1);

where as before �max = max (�1; �2; :::; �N). Then the limiting behavior of �0N�N willbe determined by N2(�max�1), namely the cross section exponent of the strongest of thedominant units, �max. For the production network to a¤ect macro economic �uctuationswe need �max > 1=2. But to make sure that �0N�N does not converge to zero as N !1,as noted earlier a much stronger condition, namely �max = 1, is required. It is clear thatthe same result follows ifW has one strongly dominant unit with � = 1. The remainingdominant units either behave similarly, or will be dominated by the leading dominantunit, with �max.Consider now Corollary 1 of Acemoglu et al. (2012), where it is established that

aggregate volatility behaves asymptotically as N�2(��1)=��2�� , for some small �� > 0 and� 2 (1; 2). Matching this rate of expansion with N2(�max�1), we have

2 (�max � 1) � �2(� � 1)=� � 2��;or �max � 1=�� ��. Therefore, �max can be viewed as measuring the inverse of �, a resultthat we formally establish below.It is also easily seen that �max � 1=&��& , where �& > 0, and & is the shape parameter of

the power law assumed by Acemoglu et al. (2012, Corollary 2) for the second-order degreesequences (based on the second-order interactions in the network). Other shape parame-ters are also considered by Acemoglu et al. (2012) for higher-order degree sequences. Butclearly these shape parameters are closely related, and are not needed since as shownabove it is possible to derive an exact rate for the asymptotic behavior of �0N�N :

16

7 Estimation and inference

In this section we consider the problem of estimating the degree of dominance of unitsin a given network. We consider the power law approach employed in the literature aswell as a new method that we propose when the outdegrees, fd1; d2; :::; dNg, follow theexponent speci�cation de�ned by (39). It is unclear if a power law speci�cation for theoutdegrees (above a given cut-o¤ value) is necessarily to be preferred to a speci�cationwhich relates the outdegrees directly to the size of the network, N , without the need tospecify a cut-o¤ value. The exponent speci�cation of outdegrees has the added advantagethat it also allows identi�cation of more than one dominant units in the network.

7.1 Power law estimators

Suppose that we have observations on the outdegrees, dit, for i = 1; 2; :::; N ; and t =1; 2; :::T . The power law estimate of �max is given by 1=�̂, where �̂ is an estimator ofthe shape parameter of the power law distribution �tted to the outdegrees that lie abovea given minimum cut-o¤ value, dmin. (See Section 6). A random variable D is said tofollow a power law distribution if its complementary cumulative density function (CCDF)satis�es

Pr (D � d) / d��; (59)

where � > 0 is a constant known as the shape parameter of the power law, and / denotesasymptotic equivalence.8 As the name suggests, the tail of the power law distributiondecays asymptotically at the power of �. It is readily seen that the probability densityfunction of D follows fD (d) / d�(�+1):A popular speci�c case of power laws is the Pareto distribution. Its CCDF is given by

Pr (D � d) = (d=dmin)�� , d � dmin; (60)

for some shape parameter � > 0 and lower bound dmin > 0. The Pareto distribution hasbeen widely used to study the heavy-tailed phenomena in many �elds including economics,�nance, geology, physics, just to name a few. Since our focus is on the estimation of theshape parameter �, in what follows we brie�y describe three approaches that are frequentlyused in the literature.9

The �rst is to run the following log-log regression (also known as Zipf regression),

ln i = a� � ln d(i), i = 1; 2; :::; Nmin; (61)

where a is a constant, i is the rank of the unit i in the sequence fd(i)g, and dmax = d(1) �d(2) � ::: � d(Nmin), are the largest ordered outdegrees such that d(Nmin) � dmin, and Nmin isthe number of cut-o¤ observations used in the regression. A bias-corrected version of thelog-log estimator of �, is proposed by Gabaix and Ibragimov (2011) who suggest shifting

8More generally, power law distributions take the form Pr (D � d) / L(d)d�� , where L(d) is someslowly varying function, which satis�es limd!1 L (rd) =L (d) = 1, for any r > 0.

9More sophisticated techniques are reviewed in, among others, Beirlant et al. (2006), and are beyondthe scope of this paper.

17

the rank i by 1=2 and estimating � by Ordinary Least Squares (OLS) using the followingregression

ln (i� 1=2) = a� � ln d(i), i = 1; 2; :::; Nmin: (62)

In what follows we consider this log-log estimator and refer to it as the Gabaix-Ibragimov(GI) estimator, which we denote by �̂GI . The standard error of �̂GI is estimated by

�̂��̂GI

�=p2=Nmin�̂GI .

Another often-used estimator of � is the maximum likelihood estimator (MLE), de-noted by �̂MLE, which is also the well-known Hill estimator (Hill, 1975). It can be easilyveri�ed that10

�̂MLE =NminPNmin

i=1 ln d(i) �Nmin ln d(Nmin); (63)

and its standard error is given by �̂��̂MLE

�= �̂MLE=

pNmin. The ML estimator is most

e¢ cient if dmin is known and the underlying distribution above the cut-o¤ point is Pareto.Finally, some researchers, notably Clauset, Shalzi and Newman (2009, CSN), pro-

pose joint estimation of � and dmin and recommend estimating dmin by minimizing theKolmogorov-Smirnov or KS statistics, which is the maximum distance between the em-pirical cumulative distribution function (CDF) of the sample, S (d), and the CDF of thereference distribution, F (d), namely,

TKS = maxd�dmin

jS (d)� F (d)j :

Here F (d) is the CDF of the Pareto distribution that best �ts the data for d � dmin. TheMLE in (63) is then computed using the estimated value of dmin. Hereafter, we refer tothis estimator as the feasible maximum likelihood estimator and denote it by �̂CSN .

11

In the subsequent analysis, we examine how the inverse of �, which is estimated bythe three procedures discussed above, behave as an estimator of �max, and how theseestimators compare to the extremum estimators that we now consider.

7.2 Extremum estimators

We now propose an extremum estimator of �max which could also be used to estimate�, and will be shown to be more generally applicable as compared to the power lawestimators of �. Our estimator is motivated by the exponent speci�cation of outdegreesgiven by (26).

7.2.1 Pure cross sections

In line with the literature on estimation of �, we begin with the case where only a singleset of observations on the outdegrees, fdig ; is available, but instead of the power law10See, for example, Appendix B of Newman (2005).11The code implementing this method can be downloaded from

http://tuvalu.santafe.edu/~aaronc/powerlaws/.

18

speci�cation we assume that di, i = 1; 2; :::; N; are generated according to the followingexponent speci�cation:

di = �N �i exp(�i); i = 1; 2; :::; N; (64)

where � > 0, �i s IID(0; �2�), and �i is distributed independently of �i. Furthermore,given the constraint (27), we must have

�

NXi=1

N �i exp(�i) = N: (65)

Taking the log transformation of (64) we have

ln di = ln�+ �i lnN + �i; i = 1; 2; :::; N:

Averaging across i now yields

N�1NXi=1

ln di = ln�+

N�1

NXi=1

�i

!lnN +N�1

NXi=1

�i: (66)

But from the summability condition, (35), it follows that N�1

NXi=1

�i

!lnN � K

�lnN

N

�;

and hence N�1

NXi=1

�i

!lnN ! 0, as N !1. (67)

Considering that by assumption �i are IID over i, then the last term of (66) also tendsto zero, and ln� can be estimated by

dln� = N�1NXi=1

ln di: (68)

An extremum estimator of �max now emerges as

�̂max =ln dmax �N�1PN

i=1 ln dilnN

=N�1PN

i=1 ln (dmax=di)

lnN; (69)

where dmax is the largest value of di > 0. As we shall see �̂max is a consistent estimator ofthe degree of pervasiveness of the most dominant unit in the network under fairly generalspeci�cations of the outdegrees, including the exponent speci�cation given by (64).The exponent speci�cation has the advantage that it is closely related to (26) in that

�j = � exp(�i) > 0, and is in line with the production network model derived from a setof underlying economic relations. Nonetheless, in practice it is di¢ cult to know if thetrue data generating process follows the exponent or a power law speci�cation. But it

19

turns out that 1=�̂max is a consistent estimator of �, the shape parameter of the Paretodistribution, even under the Pareto distribution.To see this, suppose that the observations on the outdegrees, di, for i = 1; 2; :::; N , are

independent draws from the following mixed-Pareto distribution

f(di) / d�1��i , if di � dmin; (70)

/ (di), if di < dmin;

where di > 0 follows a Pareto distribution with the shape parameter � for values of diabove dmin, and an arbitrary non-Pareto distribution, (dit), for values of di below dmin.The constants of the proportionality for both branches of the distribution are set to ensurethat

R10f(x)dx = 1, and that a given non-zero proportion of the observations fall above

dmin.Using (69), the extremum estimator, �̂max; can be rewritten as

�̂max =zmax �N�1PN

i=1 zilnN

; (71)

where zi = ln(di=dmin), for all i, and z(i) = ln(d(i)=dmin), with d(i) being the ith largest valueof di as before. Since dmin is a given constant and by assumption di are independentlydistributed, it then follows that for zi � 0, zi are independent draws from an exponentialdistribution with parameter �, namely

fZ(z) = �e��z, for z � 0,

with E (z jz � 0) = 1=�, and V ar (z jz � 0) = 1=�2, for � > 0.12 We also assume thatE (zi jzi < 0) exists, which is a mild moment condition imposed on (di) for ln (di=dmin) <0. The following proposition summarizes the consistency property of �̂max as an estimatorof 1=�.

Proposition 3. Suppose that di, for i = 1; 2; :::; N , are independent draws from thePareto tail distribution given by (70) with the shape parameter � > 0, and assume thatzi = ln(di=dmin) has �nite second order moments for all values of zi < 0. It then followsthat

limN!1

E��̂max

�= 1=�, and V ar

��̂max

�= O

�1

(lnN)2

�; (72)

where �̂max is de�ned by (71).

A proof is provided in Appendix C.

Remark 4. The convergence of �̂max to � = 1=�, is at the rate of 1= lnN which is ratherslow. But it is obtained without making any assumptions about dmin and/or the shape of (d), the non-Pareto part of the distribution.

12It is worth noting that z has moments even if � � 1, although the Pareto distribution has momentsonly for � > 1.

20

7.2.2 Short T panels

Suppose now that observations on outdegrees, dit, are available across t = 1; 2; :::; T andare generated as before, namely

dit = �N �i exp(�it); i = 1; 2; :::; N ; t = 1; 2; :::; T; (73)

where T is �nite (T > 1) and N is large, � > 0, and �i � 0 are �xed constants. �it sIID(0; �2�) over i and t.

13 Noting that (65) must hold and using a similar line of reasoningas in the pure cross section case we obtain the following "panel extremum estimator"

�̂max =T�1

PTt=1 ln dmax;t � (TN)

�1PTt=1

PNj=1 ln djt

lnN; (74)

where dmax;t is the largest value of dit for period t. Also for other units we have

�̂i =T�1

PTt=1 ln dit � (TN)

�1PTt=1

PNj=1 ln djt

lnN; (75)

where �̂i is the estimator of �i. Recall that only estimates of �̂i > 1=2 should be considered,since any unit with an estimate of �i below 1=2 will not have any network e¤ects (seeRemark 3). We denote the degree of dominance of unit i by �i, and the associated orderedvalues by �(i), where �max = �(1) � �(2) � ::: � �(N): Then the second largest �̂i, denotedby �̂(2); is a consistent estimator of �(2); the third largest �̂i, denoted by �̂(3); is a consistentestimator of �(3); and so on. We refer to �̂(i), for i = 1; 2; :::;m, as the extrema estimators,with �̂(m) > 1=2.To investigate the asymptotic properties of the extrema estimators, we �rst note that

under (73),dln� and �̂i can be written asdln�� ln� = �� lnN + ��; (76)

�̂i � �i = �� +��i � ��lnN

; (77)

where dln� = (TN)�1PT

t=1

PNi=1 ln dit, �� = N�1PN

i=1 �i; ��i = T�1PT

t=1 �it, and �� =N�1PN

i=1 ��i: It is now easily seen that

Cov(�̂i; �̂j) = � 1

(lnN)2N

�2�T, for all i 6= j;

V ar��̂i

�=

�2�(lnN)2 T

�1� 1

N

�: (78)

Clearly for any given i, V ar��̂i

�! 0; for a �xed T asN !1, and the rate of convergence

of V ar��̂i

�is given by (lnN)�2. Also, it is already established that �� ! 0, as N ! 1,

13This assumption can be relaxed to allow for both heteroskedasticity and serial correlation if T issu¢ ciently large. This extension is considered in an Online Supplement.

21

and as a result �̂i !p �i, which establishes that for any �nite T � 1, �̂i is a lnN consistentestimator of �i. The estimators of � for two di¤erent units (i; j), are asymptoticallyindependent and their covariance converges to zero at the faster rate of (lnN)�2N�1.Then, for any given i we have (applying standard central limit theorem to ��i � ��)�

�̂i � �i � ���

h�2�

(lnN)2T

�1� 1

N

�i1=2 !d N(0; 1), as N !1:

Ignoring lower order terms in N we now obtain

(lnN)pT��̂i � �i � ��

���

!d N(0; 1), as N !1:

To ensure that the above statistic does not depend on the nuisance parameter, ��, we need

�� (lnN)pT =

NXi=1

�i

!(lnN)

pT

N! 0, as N !1: (79)

But given the summability condition, (35), it is clear that this condition is met if T is�xed, as N !1.To obtain a consistent estimator of �2�, note that

�̂it = ln dit �dln�� �̂i lnN (80)

= ��dln�� ln��� ��̂i � �i

�lnN + �it:

Now using (76) and (77)

�̂it = �2�� lnN + �it � ��i;

= �it � ��i +O

�lnN

N

�:

In view of this result, �2� can be consistently estimated by (for T > 1)

�̂2� =

PNi=1

PTt=1 �̂

2it

N (T � 1) : (81)

A test of the null hypothesis that �max = �0max, where �0max > 1=2, can be based on the

statistic

Dmax =(lnN)

��̂max � �0max

��̂��1T� 1

TN

�1=2 : (82)

It then follows that Dmax !d N(0; 1); as N !1, for a �xed T > 1.

22

7.2.3 Unbalanced panels

In empirical applications, production networks observed at di¤erent points in time mightnot have the same units in common. As a result we are often faced with unbalanced paneldata sets. One approach would be to employ a su¢ ciently high level of aggregation sothat we end up with a balanced panel. But this procedure is likely to be ine¢ cient as weend up with a smaller number of units in the network. Here we consider estimating �iwith the unbalanced panel, without any aggregation. We suppose for each unit i we haveobservations on its outdegrees for at least two time periods.We denote the unbalanced panel of observations on the outdegrees by dit, for t =

T 0i ; T0i + 1; :::; T

1i , (T

1i � T 0i ), and i = 1; :::; N . Then using a similar line of reasoning as

above we have

�̂i =T�1i

PT 1it=T 0i

ln dit �N�1PNi=1

�T�1i

PT 1it=T 0i

ln dit

�lnN

; (83)

where Ti = T 1i � T 0i + 1, and

V ar��̂i

�=

�2�(lnN)2

�1

Ti� 1

NTi

�: (84)

The estimator of the most dominant unit is given by �̂max, and as in the balanced panelcase, the asymptotic distribution of �̂max is given by

Dmax =(lnN)

��̂max � �0max

��̂�

�1

Tmax� 1

NTmax

�1=2 ; (85)

where Tmax refers to the sample size of the most dominant unit, and

�̂2� =

PNi=1 (Ti � 1)

�1PT 1it=T 0i

�̂2it

N: (86)

The distribution of the most dominant unit is well de�ned if Tmax > 1.

7.2.4 Comparison of power law and extremum estimators

As compared to the power law estimators, the extremum estimator has several advantages.First, it does not require knowing the true value of dmin, whereas the estimates of the shapeparameter may be highly sensitive to the choice of the cut-o¤ value. Although proceduressuch as the feasible MLE proposed by Clauset et al. (2009) estimate dmin jointly with �,such estimates assume that the true distribution below and above dmin are known, whilstthe extremum estimator is robust to any distributional assumptions below dmin, so long asln(di=dmin) has second order moments. Granted that it may not be as e¢ cient as MLE ifthe true distribution is indeed Pareto, one does not need to make such strong assumptionson the entire distribution. Third, the extremum type estimators can identify the dominantunits besides the most dominant one, and estimate the degrees of pervasiveness of eachof the of the dominant units separately.

23

8 Monte Carlo experiments

In this section, we investigate the small sample properties of the proposed extremumestimator for balanced panels using Monte Carlo techniques, and compare its performancewith that of the power law method.14

We consider two types of data generating processes (DGPs) for the outdegrees (dit):an exponent speci�cation and a power law speci�cation. The DGP for the exponentspeci�cation is given by

ln dit = ln�+ �i lnN + �it; i = 1; 2; :::; N ; t = 1; 2; :::; T; (87)

where �it � IIDN(0; 1), with � set as

� =exp

��12

�N�1PN

i=1N�i> 0; (88)

to ensure that dit add up to N across i for each t.For the power law model we closely follow Clauset et al. (2009), and initially generate

yit as random draws from the following mixture distribution that obeys an exact Paretodistribution above ymin;t and an exponential distribution below ymin;t:

f(yit) =

8><>:Ct(yit=ymin;t)

�(�+1); for yit � ymin;t

Cte�(�+1)(yit=ymin;t�1); for yit < ymin;t

; (89)

for i = 1; 2; :::; N , and t = 1; 2; :::; T . To ensure that f(yit) integrates to 1 over its fullsupport, yit > 0, we set Ct as

Ct =

"ymin;t

�e�+1 � 1

�� + 1

+ymin;t�

#�1: (90)

We then set dit = yit=�yt and dmin;t = ymin;t=�yt, where �yt = N�1PNi=1 yit, which ensure that

the outdegrees add up to N . It is worth noting that under this DGP

Pr (dit � dmin;t) = Pr (yit � ymin;t) =1

�

�e�+1 � 1� + 1

+1

�

��1; (91)

which is time-invariant and depends only on the value of �.15 The inverse transformationsampling method is used to generate yit such that its distribution satis�es (89). To thisend we �rst generate uit as IIDU [0; 1], i = 1; 2; :::; N ; t = 1; 2; :::; T; and set

umin;t = Ct

�ymin;t� + 1

��e�+1 � 1

�; (92)

14Small sample properties of the extremum estimator for unbalanced panels are investigated in theOnline Supplement.15When T > 1, we construct a panel data assuming that all units maintain their relative dominance

over time, and therefore for each t we sort dit in a descending order.

24

and then generate yit as

yit =

8>>>>>><>>>>>>:

� ymin;t� + 1

ln

�1� (� + 1)uit

Cte�+1ymin;t

�, if uit < umin;t

"� (ymin;t � uit) + Ctymin;t

Cty�+1min;t

#�1=�, if uit � umin;t

: (93)

We carry out two sets of experiments based on the above two DGPs:Exponent DGP: Observations on dit are generated according to the exponent speci-

�cation, (87), with a �nite number of dominant units and a large number of non-dominantunits. Speci�cally

� A.1. One strongly dominant unit: �max = �(1) = 1; with �(i) = 0 for i = 2; 3; :::; N:

� A.2. Two strongly dominant units: �max = �(1) = �(2) = 1; with �(i) = 0 fori = 3; 4; :::; N:

� A.3. One strongly dominant unit and one weakly dominant unit: �max = �(1) = 1and �(2) = 0:75, with �(i) = 0 for i = 3; 4; :::; N .

We consider all combinations of N = 100; 300; 500; and 1; 000, and T = 1; 2; 6; 10; and20, and also provide simulation results for a very large data set with N = 450; 000, whichcan arise when using inter-�rm level sales data.16 We focus on the 5 largest estimates of�, which we denote by �̂max = �̂(1) � �̂(2) � ::: � �̂(5), computed according to (75). WhenT > 1, the variance of �̂(i) is computed by (78), with �2� estimated by (81).Pareto DGP: Observations on dit are generated according to the mixture Pareto

distribution, (89), described above and we consider Experiments B.1: � = 1:0, and B.2:� = 1:3. The values of ymin;t are set as ymin;t = ymin = 15. The sample sizes arecombinations of N = 100; 300; 500; 1; 000; and 450; 000; and T = 1 and 2. We assessthe performance of the Gabaix-Ibragimov estimator (�̂GI) given by (62) for di¤erentgiven cut-o¤ values, dmin;t, the maximum likelihood estimator (�̂MLE) given by (63) fordi¤erent dmin;t, and the CSN estimator (�̂CSN) which estimates � jointly with the cut-o¤value.As shown in Proposition 3, the inverse of the extremum estimator, 1=�̂max, is a con-

sistent estimator of �, and analogously one can consider the inverse of � as an estimator�max.17 It is therefore of interest to see how the extremum estimator performs under thePareto DGP, and conversely how the power law estimators perform under the ExponentDGP. To investigate the robustness of the alternative estimators of � to the choice ofthe underlying distribution, we conduct two sets of misspeci�cation experiments wherewe compare the small sample properties of the four estimators of �, namely �̂GI , �̂MLE,

16For example, Carvalho et al. (2016) use a subset of data compiled by Tokyo Shoko Research Ltd thatcontains information on inter-�rm transactions of around one million �rms across Japan. This data setis proprietary and has not been made available to us.17See also the discussions in Section 6 on the relationship between �max and �.

25

�̂CSN ; and �̂max = 1=�̂max; when the underlying DGP is Pareto, and conversely whenExponent DGP is used. We consider the values of � = 1:0, and 1:3 under Pareto DGP,and �max = 1 and 1=1:3 under Exponent DGP. We focus on small values of T = 1 and 2,for all combinations of N = 100; 300; 500; 1; 000; and 450; 000.All experiments are carried out with 2; 000 replications.18

8.1 MC results

The estimation results under Exponent DGP for Experiments A.1 to A.3 are summarizedin Table 1, and focus on the extremum estimators of �max = �(1) and �(2) when applicable.For each experiment we report bias (�100), root mean squared error (RMSE�100), aswell as size (�100) and power (�100) for the estimators under consideration. We �rst notethat the bias and RMSE of the extremum estimators decline as N and/or T rises. Thebias and RMSE reduction is particularly pronounced as T is increased. This is in line withthe theoretical derivations which establish that along the cross-sectional dimension therate of convergence is of order 1= ln(N), as compared to T�1=2 along the time dimension.We also note that the empirical sizes of the tests based on �̂max and �̂(2) are close to theassumed 5% nominal size in most cases. There is some over-rejections in cases where Tis much larger than N; and when there are more than one dominant units. In practice,this is unlikely to be a real concern since N is typically much larger than T . Seen fromthis perspective, it is particularly satisfying to note that the extremum estimator hassatisfactory performance even when N approaches 450; 000. The slow rate of convergencealong the cross section dimension is, however, important for the power of the test. Forexample, in the case of Experiment A.1, the power of detecting the strongly dominantunit (against the alternative that �max = 0:90) is around 9% for N = 100 and T = 2,and rises only slowly as N is increased. However, we see a signi�cant rise in power if T isincreased to 6. For T = 6 the power rises from 17% for N = 100 to 89% for N = 450; 000,twice as much as the values obtained for T = 2. The power also rises as the number ofstrongly dominant units is raised from one to two.We also consider the frequency with which the dominant units are jointly selected

under Exponent DGP, Experiments A.1 to A.3. The results are summarized in Table 2,and show that units with � equal to unity are almost always correctly selected, especiallywhen T > 2. The selection frequency can be low in the case of Experiment A.3 wherethe network has two dominant units one of which is weak with � = 0:75. In such caseswe need N to be quite large if T is less than 3.Tables 3 and 4 summarize the results for the �rst set of misspeci�cation experiments

where the data are generated from the Pareto tail distribution given by (70). For di¤erentvalues of �, the extremum estimator demonstrates robustness to the model misspeci�ca-tion, although it converges to the true value much more slowly than the other shapeestimators under Pareto type distributions. This �nding is in line with our theoreticalresults provided in 7.2.1. The extremum estimator, �̂max = 1=�̂max, performs well particu-larly when � = 1, even when N is relatively small. For example, under Pareto DGP with

18We also investigate the small sample properties of the extremum estimator of �max for three othersets of experiments: (a) exponentially decaying �i�s, (b) unbalanced panels, and (c) heteroskedastic andserially correlated errors in the case of large N and large T panels.

26

� = 1 (Experiment B.1), for N = 300 and T = 2, �̂max = 1:01 (0:05); while �̂GI = 1:04(0:19) and �̂MLE = 1:05 (0:14), assuming a 10% cut-o¤ value.19 It is also worth notingthat the Gabaix-Ibragimov estimator (�̂GI) and the ML estimator (�̂MLE) are quite sen-sitive to the choice of the cut-o¤ values.20 The feasible MLE, �̂CSN ; performs better, butit is important to note that the validity of the feasible MLE procedure critically dependson how close the assumed speci�cation of the distribution of dit below dmin;t is to the trueunderlying distribution.Consider now the case where the outdegrees are generated according to the exponent

speci�cation, (87). In this misspeci�ed case the Pareto estimators, �̂GI , �̂MLE, and �̂CSN ;all show signi�cant biases (see Tables 5 and 6). For instance, when � = 1, N = 300 andT = 2, and the outdegrees are generated according to the Exponent DGP, the bias of�̂GI ranges from 0:10 to 0:35, for the cut-o¤ values 10% to 30%. Also, the bias of �̂GIincreases rapidly with N . The ML type estimators exhibit similar biases.Finally, the extremum estimator continues to perform well in the case of unbalanced

panels, and large N and T panels with heteroskedastic and serially correlated errors. It isalso reasonably robust to alternative network structures under di¤erent speci�cations ofthe distribution of outdegrees, such as exponentially decaying �i�s. The relevant summarytables are available in the Online Supplement.

9 Dominant units in US production networks

In this section we apply the proposed estimation strategy to identify the top �ve mostpervasive (dominant) sectors in the US economy. We also compare our results with theestimates of � (the inverse of �max) obtained by Acemoglu et al. (2012) for the mostdominant sector. We provide estimates based on the US input-output tables for singleyears as well as when two or more input-output tables are pooled in an unbalanced panel.Acemoglu et al. (2012) only consider the estimates of � based on single-year input-outputtables.We begin with a re-examination of the data set used by Acemoglu et al. (2012) so that

we have a direct comparison of the estimates of � (or its inverse) based on the shape ofthe power law, and the extremum estimates which is given by �̂max, and computed using(74). The Acemoglu et al. (2012) data set are based on the US input-output accounts dataover the period 1972-2002 compiled by the Bureau of Economic Analysis (BEA) every�ve years. We �rst con�rmed that we can replicate their estimates of �, which we denoteby �̂GI assuming a 20% cut-o¤ value (the percentage above which the degree sequencesare assumed to follow the Pareto distribution). The estimates �̂max and the inverse of �̂for the years 1972, 1977, 1982, 1987, 1992, 1997 and 2002 are given in Tables 7 and 8. Forthe inverse of �̂, Tables 7 and 8 report estimates based on the �rst-order and second-orderinterconnections, respectively.21 We estimate � by the three approaches discussed above,

19Figures in brackets are standard errors.20Similar Monte Carlo evidence illustrating the truncation sensitivity problem is reported in Table 1-4

of Gabaix and Ibragimov (2011). An interesting theoretical discussion can be found in Eeckhout (2004).21The �rst-order degree of sector j is just its outdegree, dj , de�ned as before, while the second-order

degree of sector j is de�ned by dj;2 = d0w�j , where d = (d1; d2; :::; dN )0 is the vector of �rst-order degrees

27

namely Gabaix-Ibragimov estimator (�̂GI) given by (62), the MLE (�̂MLE) given by (63),and the feasible MLE (�̂CSN). For the Gabaix-Ibragimov regression and MLE, we giveestimates for the cut-o¤ values of 10%, 20%, and 30%. For the feasible MLE, we presentboth the estimates of � and the estimated cut-o¤ values.22

The results in Tables 7 and 8 show that the yearly estimates of �max are clusteredwithin the narrow range of 0:77 to 0:82, covering a relatively long period of 30 years. Wecan not provide standard errors for such yearly estimates, but given the small over-timevariations in these estimates we can con�dently conclude that there is a high degree ofsectoral pervasiveness in the US economy, although these estimates do not support thepresence of a strongly dominant unit which requires �̂max to be close to unity. In contrast,the estimates of �max based on the inverse of �̂ di¤er considerably depending on theestimation methods, the choice of the cut-o¤ value, and whether the �rst- or second-orderinterconnections are considered. For example, for 1972, the estimates based on the powerlaw, inverse of �̂GI , range from 0:694 when the cut-o¤ value is 10% and the �rst-orderinterconnections are used, and rise to 1:035 when the second-order interconnections areused with a 30% cut-o¤ value. The estimates of � based on the inverses of �̂GI and�̂MLE, rise with the choice of cut-o¤ values and with the order of interconnections, whilstour estimator does not require making such choices. Recall that �max provides an exactmeasure of the rate at which the variance of aggregate output responds to sectoral shocks,whilst � characterizes a lower bound if the �rst-order interconnections are used. A 20%cut-o¤ value, which is assumed by Acemoglu et al. (2012) seems reasonable, consideringthe closeness between the estimates of �̂max and the inverse of �̂GI , and given its similarityto the estimated cut-o¤ values by the feasible MLE. Nevertheless, the estimated cut-o¤value based on the �rst-order interconnections for the year 1992 is only 9:5%, which ismarkedly lower than that for the other years. Similar issues arise when the second-orderinterconnections are used. The di¤erences between �̂max and inverse of �̂GI also vary acrossthe years. For example, using the second-order interconnections and a cut-o¤ value of20%, �̂max and inverse of �̂GI are reasonably close for the years 1992, 1997 and 2002, butdiverge for the earlier years of 1972, 1977 and 1982.The data sets provided by Acemoglu et al. (2012) do not give the identities of the

sectors, which is �ne if one is only interested in � or �max. But, as noted earlier, ourestimation approach also allows us to identify the sectors with the highest degrees ofpervasiveness in the production network. With this in mind, we compiled our own Wmatrices from the input-output tables downloaded from the BEA website.23 The top �ve

and w�j is the jth column ofW.22Acemoglu et al. (2012) estimated the shape parameter of the power law by the log-log regression

and non-parametric Nadaraya-Watson regression, taking the tail to correspond to the top 20% of thesamples for each year and did not try other cut-o¤ values. They also estimated the shape parameterby the feasible maximum likelihood method proposed by Clauset et al. (2009), but did not report theestimates for each year or the estimated cut-o¤ values.23The W matrices for di¤erent years were computed from commodity-by-commodity direct require-

ments tables at the detailed level that cover around 400-500 US industries. The (i; j)th entry of such atable shows the expenditure on commodity j per dollar of production of commodity i. As in Acemogluet al. (2012), the terms sector and commodity are used interchangeably to convey the same meaning.These direct requirements tables can be derived from the total requirement tables at the detailed level,which are compiled by the BEA every �ve years. Further details on the data description and transfor-

28

largest estimates of �, denoted by �̂max = �̂(1) � �̂(2) � ::: � �̂(5), for each of the years1972 to 2007 are given in Table 9. The identities of the associated sectors are given inTable 10. We note that both the degrees of dominance and the identities of the pervasivesectors in the US economy are relatively stable over the years. Consistent with the resultsin Table 7, no sector is strongly dominant. The highest estimate of �max is 0:82, for theyear 1992, with an average estimate of around 0:78 over the sample. The wholesale tradesector turns out to be the most dominant sector for all the years with the exception of2002. In this year the management of companies and enterprises is the most dominantsector with the wholesale trade coming second.But it is generally di¢ cult to distinguish between the top two or three sectors as their

� estimates are quite close to one another and we are not able to apply formal statisticaltests to their di¤erences as standard errors can not be computed using outdegrees for onesingle year.24 Accordingly, to provide more reliable estimates of �(1); �(2); :::; �(5) and theassociated sectoral identities, we also consider pooled estimates. However, there have beenmajor changes in the BEA industry classi�cations over the years, with the input-outputtables for the period 1972-1992 being based on the Standard Industrial Classi�cation(SIC) system, while starting from 1997 they are based on the North American IndustryClassi�cation System (NAICS). Accordingly, we computed panel estimates of � for thetwo sub-samples separately. The results are summarized in Table 11, which also givesstandard errors in parentheses, computed using (84). It is interesting that despite changesto the sectoral classi�cations, the wholesale trade sector is identi�ed as the most dominantsector in both sub-samples, with �̂max = 0:762 (0:036) for the �rst sub-sample (1972-1992), and �̂max = 0:716 (0:045) for the second sub-sample (1997-2007). The two panelestimates are quite close and identify wholesale trade as the most dominant sector in theUS economy. Turning to the estimates of �(2); �(3); :::; �(5), we �nd that these estimatesare also remarkably similar across the two sub-samples, ranging from 0:667 to 0:605 inthe �rst sub-sample, and 0:683 to 0:595 in the second sub-sample. What has changedis the identity of the sectors across the two sub-samples. For example, the second mostdominant sector has been blast furnaces and steel mills over the �rst sub-sample (1972-1992), whilst it is management companies and enterprises over the second sub-sample(1997-2007).

10 Concluding remarks

This paper extends the production network considered by Acemoglu et al. (2012) andderives a dual price network, which allows us to obtain exact conditions under whichsectoral shocks can have aggregate e¤ects. The paper presents a simple nonparametricestimator of the degree of pervasiveness of sectoral shocks that compares favorably withthe parametric estimators based on Pareto distribution �tted to the outdegrees. Theproposed extremum estimator is simple to implement and is applicable not only to thepure cross section models where the Pareto shape parameter is estimated, but also extends

mations can be found in Appendix D.24Acemoglu et al. (2012) are able to compute standard errors for their estimates of � because they

impose a Pareto distribution on the ordered outdegrees beyond a cut-o¤ point, which they assume.

29

readily to short T as well as large T panels. The paper also develops a simple test ofthe degree of pervasiveness of the most dominant units in the network, which is shownto have satisfactory size and power properties when N is large, even if T is quite small.The production and price networks considered in this paper are static, but the proposedstatistical framework can be extended to allow for dynamics, along similar lines as inPesaran and Chudik (2014) who consider aggregation of large dynamic panels.Our empirical application to US input-output tables suggests some evidence of sector-

speci�c shock propagation, but such e¤ects do not seem su¢ ciently strong and long-lasting, and are likely to be dominated by common technological e¤ects. Similar empiricalevidence are also provided by Foerster, Sarte, andWatson (2011), who incorporate sectorallinkages into multisector growth models producing an approximate factor model. Theirfactor analytic approach, however, cannot distinguish dominant unit(s) from commonfactors and therefore may underestimate the in�uence of input-output linkages.25 Theissue of the relative importance of internal network interactions and external commonshocks for macro economic �uctuations continues to be an open empirical question.

A Appendix: Lemmas

Lemma 1. Let A be an N �N matrix whose entries are non-negative and each row sums upto 1. Then �1 (A) = 1, where �1 (A) is the largest eigenvalue of A, and IN � �A is invertiblegiven that j�j < 1.

Proof. Matrix A is as a right stochastic matrix, and �1 (A) = 1 follows. See, for example,Property 10.1.2 in Stewart (2009). Given that j�j < 1 and �1 (A) = 1, it then readily seen thatall eigenvalues of IN � �A are strictly positive in absolute value, and hence invertible.

Remark 5. It should be noted that this lemma holds irrespective of whether A has boundedcolumn matrix norm. Also note that �1 (A0) = 1 and IN � �A0 is invertible, since a matrix andits transpose always have the same set of eigenvalues.

Lemma 2. Let A be an N �N matrix and B = IN � �A: Suppose that

j�j < max (1= kAk1 ; 1= kAk1) :

Then B�1 has bounded row and column sum matrix norms.

Proof. See Pesaran (2015, p.756).

B Appendix: Multiple dominant unitsThis appendix extends the analysis of Section 5 to the scenario where there are more than onedominant unit in the network. Speci�cally, we assume that the �rst m units are dominant withdegrees of dominance f�1;�2; :::; �mg, and the rest n units are non-dominant, with �i = 0, for

25The factor analysis also requires large N and T panels and is not applicable when T is small.

30

i = m + 1;m + 2; ::::;m + n, and let N = m + n. Consider now the following partitionedversion of model (24)�

x1tx2t

�=

��W11 �W12

�W21 �W22

��x1tx2t

�+

�g1tg2t

�;

where x1t = (x1t; x2t; :::; xmt)0, x2t = (xm+1;t; xm+2;t; :::; xNt)0,W11 is the m�m weight matrixassociated with the dominant units, W22 is the n � n weight matrix associated with the non-dominant units and assumed to satisfy j�j kW22k1 < 1, and g1t = (g1t; g2t; :::; gmt)

0, g2t =(gm+1;t; gm+2;t; :::; gNt)

0, where git = �bi � �( ift + "it); for i = 1; 2; :::; N . As %(W22) � 1 andj�j < 1, we have

x2t = S�122 (�W21x1t + g2t) ; (B.1)

where S22 = In � �W22. Substituting (B.1) into

x1t = �W11x1t + �W12x2t + g1t;

and rearranging yieldsx1t = Z

�11 g1t + �Z

�11 W12S

�122 g2t; (B.2)

whereZ1 = Im � �W11 � �2W12S

�122W21; (B.3)

and Z1 is invertible as (IN � �W) is nonsingular by Lemma 1 in Appendix A.Now consider the cross-section average of xit for i = 1; 2; :::; N ,

xNt = N�1 �� 0mx1t + � 0nx2t� :Using (B.1) in the above equation gives

xNt = N�1 ��� 0m + �� 0nS�122W21

�x1t + �

0nS

�122 g2t

�;

and by the de�nition of g1t we obtain

xNt = N�1 ��an + �0nx1t � � nft � ��0n"2t� ;where an = � 0nS

�122 b2; �

0n = �

0m+��

0nW21; n = �

0n 2; �

0n = �

0nS

�122 ; with b2 = (bm+1; bm+2; :::; bN )

0

and 2 =� m+1; m+2; :::; N

�0.We will derive V ar(xNt) and inspect its asymptotic order of magnitude as N ! 1 fol-

lowing similar steps as in Section 5. First, as with the case of the SAR(1) model with onedominant unit, we have 1 < �min � �max < K < 1, where �0n = (�m+1; �m+2; :::; �N );�min = min(�m+1; �m+2; :::; �N ); and �max = max(�m+1; �m+2; :::; �N ): Also, it readily fol-lows that an = O(1) and V ar

�N�1�0n"2t

�=

�N�1� : Considering now the terms due to the

dominant units, and using (B.2) note that

Cov�x1t; N

�1�0n"2t�= Cov

�Z�11 g1t + �Z

�11 W12S

�122 g2t; N

�1�0n"2t�

(B.4)

= �N�1��Z�11 W12S�122 V22;"S

�1022 �n;

Cov (x1t; ft) = ��V ar(ft)�Z�11 1 + �Z

�11 W12S

�122 1

�; (B.5)

V ar (x1t) = �2Z�11 V11;"Z0�11 + �2�2Z�11 W12S

�122 V22;"S

�1022 W

012Z

0�11 (B.6)

+�2V ar(ft)�Z�11 1

01Z

0�11 + �2Z�11 W12S

�122 2

02S�1022 W

012Z

0�11

�;

31

whereV11;" = diag��21; �

22; :::; �

2m

�;V22;" = diag

��2m+1; �

2m+2; :::; �

2N

�, and 1 = ( 1; 2; :::; m)

0.

Consider now the individual terms of V ar(xNt); which is given by

V ar (xNt) = N�2�0nV ar(x1t)�n � 2�N�2�0nCov�x1t;�

0n"2t

�+ �2N�2V ar

��0n"2t

�+�2N�2V ar(ft)

� 2n + 2 n�

0nZ

�11 1 + 2� n�

0nZ

�11 W12S

�122 2

�:

In the case where the network contains m dominant units but is not subject to common shocks,

V ar (xNt) = N�2�0nV ar(x1t)�n � 2�N�2�0nCov�x1t;�

0n"2t

�+(N�1): (B.7)

Consider the ith element of N�1�n, denoted by N�1�i;n, for i = 1; 2; :::;m, and note that m is�xed and does not rise with N . Then by de�nition, N�1�i;n= N�1 (1 + ��0nw�i;21) ; wherew�i;21 is the ith column ofW21. Hence

�minN�1

nXj=1

wji;21 � N�1�0nw�i;21 � �maxN�1

nXj=1

wji;21;

and

N�1 + �minN�1

nXj=1

wji;21 � N�1�i;n � N�1 + �maxN�1

nXj=1

wji;21: (B.8)

Also note that w0�i;21�n = (N �i), which immediately follows that

w0�i;21�n +w0�i;11�m = di = �iN

�i ;

with m being �xed. Therefore, by (B.8) it follows that N�1�i;n = (N �i�1); for i = 1; 2; :::;m;and then N�2�0n�n = (N2�max�2); where 0 < �max = max (�1; �2; :::; �m) � 1. Further noticethat

N�2�0n�n�m [V ar(x1t)] � N�2�0nV ar(x1t)�n � N�2�0n�n�1 [V ar(x1t)] ;

where �1 [V ar(x1t)] and �m [V ar(x1t)] denote the largest and smallest eigenvalue of V ar(x1t),respectively, and 0 < �m [V ar(x1t)] � �1 [V ar(x1t)] < K <1. Hence we obtain

N�2�0nV ar(x1t)�n = (N2�max�2):

Turning now to the jth element of the covariance term, for j = 1; 2; :::;m, we have��Cov �xjt; N�1�0n"2t��� � N�1 j��j

Z�1j:;1 1 kW12k1 S�122 1 kV22;"k1 k�nk1 ;

where Z�1j:;1 denotes the jth row of Z�11 , and using similar line of reasoning as in the main text

it is easily veri�ed that jCov (xjt; N�1�0n"2t)j = O (N�1) ; and therefore

N�2�0nCov (x1t;�0n"2t) = O(N �max�2):

Consequently, in the absence of common shocks, we have demonstrated that

V ar (xNt) = (N2�max�2) +(N�1);

which clearly shows that the rate of convergence of xNt depends on the strongest dominant unitin the network.

32

Finally, if the network is subject to both dominant units and a common factor, it immediatelyfollows from N�1 n = (N � �1) and similar arguments as before that

V ar (xNt) = (N2�max�2) +(N2� �2) +(N�1);

which is a direct extension of (58) to the multiple-dominant-units network, and shows that therelative magnitudes between �max and � determines the limiting properties of the aggregatee¤ects.

C Appendix: Consistency of �̂maxFirst we note that since zi are distributed independently with �nite means and variances then

E (�zN ) = N�1NXi=1

E (zi) = ��1 Pr (z � 0) + E (z jz < 0) [1� Pr (z � 0)] ;

which is �nite. Further using standard results for the moments of ordered random variables(see, for example, Section 4.6 of Arnold et al. (1992)) we have

E�z(i)�= (1=�)

0@N�i+1Xj=1

1

j

1A ; V ar(z(i)) = (1=�)2

0@N�i+1Xj=1

1

j2

1A , for i = 1; 2; :::; N: (C.1)

Taking expectations and variance of �̂max given by (71), and making use of the above results wenow have

E��̂max

�=E (zmax)� E (�zN )

lnN=(1=�)

PNj=1 j

�1 � E (�zN )lnN

; (C.2)

V ar��̂max

�=

V ar (zmax) +N�2PN

i=1 V ar(zi)� 2N�1PNi=1Cov(zmax; z(i))

(lnN)2

=V ar (zmax) +N

�2PNi=1 V ar(zi)� 2N�1PN

i=1 V ar(z(i))

(lnN)2: (C.3)

Also using well known bounds to harmonic series (see, for example, Section 3.1 and 3.2 of Bonaret al. (2006)), we have

ln(N + 1) <

0@ nXj=1

1

j

1A � 1 + lnN;

and hence

limN!1

PNj=1 j

�1

lnN= 1: (C.4)

Using (C.1) and (C.4) in (C.2) we now have limN!1E��̂max

�= 1=�:

33

Turning to the variance of �̂max, we note that

V ar��̂max

�=

V ar (zmax) +�1�2NN2

�PNi=1 V ar(z(i))

(lnN)2;

(lnN)�2 V ar (zmax) � V ar��̂max

�� (lnN)�2

�V ar (zmax) +

�2N � 1N2

�NV ar (zmax)

�;

(lnN)�2 �2

0@ NXj=1

1

j2

1A � V ar��̂max

�� (lnN)�2

�3N � 1N

��2

0@ NXj=1

1

j2

1A :

ButPNj=1 j

�2 � �2=6, and hence V ar��̂max

�= O

h(lnN)�2

i.

D Data appendixThe input-output accounts data are obtained from the Bureau of Economic Analysis (BEA)website at http://www.bea.gov/industry/io_annual.htm. The input-output tables at the �nestlevel of disaggregation are compiled every �ve years, and the latest available data are for year2007. We derive the Commodity-by-Commodity Direct Requirements (DR) table by applyingthe following formula:

DR = (TR� I) (TR)�1 ;

where I is an identity matrix, and TR denotes the Commodity-by-Commodity Total Require-ments table that is available from the BEA. The input-output matrix,W, is set to the transposeof DR and row-standardized so that the intermediate input shares sum to one for each sector.The sectors without any direct requirements and those with zero outdegrees are excluded fromW.

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Bai, J. and K. Li (2013). Spatial panel data models with common shocks. Available at SSRN2373628.

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36

Table1:Bias,RMSE,sizeandpoweroftheextremum

estimatorforthedominantunitorunitsunderExponentDGPfor

ExperimentsA.1toA.3

Bias(�100)

RMSE(�100)

Size(�100)

Power(�100)

TnN

100

300

500

1,000

450,000

100

300

500

1,000

450,000

100

300

500

1,000

450,000

100

300

500

1,000

450,000

ExperimentA.1:Onestronglydominantunit,� m

ax=1

1-0.90

-0.46

-0.31

-0.21

-0.06

20.87

17.28

15.89

14.34

7.63

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

2-1.29

-0.55

-0.40

-0.29

-0.10

15.31

12.40

11.37

10.24

5.44

5.75

5.25

4.90

5.05

4.95

8.90

11.55

12.75

14.85

44.45

6-1.19

-0.49

-0.35

-0.23

-0.07

8.83

7.09

6.51

5.85

3.10

4.75

4.60

4.60

4.60

4.65

16.70

25.45

30.40

37.40

89.10

10-1.26

-0.55

-0.39

-0.27

-0.09

6.98

5.61

5.14

4.62

2.45

5.35

5.35

5.15

5.20

5.35

24.45

39.10

46.55

56.35

98.75

20-1.14

-0.44

-0.29

-0.19

-0.04

4.97

3.94

3.61

3.24

1.72

5.25

4.80

4.90

4.95

4.90

45.65

67.90

76.85

86.10

100.00

ExperimentA.2:Twostronglydominantunits,� (1)=� (2)=1

� (1)=1

19.89

8.91

8.39

7.72

4.20

20.38

16.99

15.73

14.25

7.63

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

26.15

5.93

5.65

5.24

2.89

13.99

11.78

10.92

9.91

5.32

4.00

4.45

4.25

4.45

4.75

14.95

20.25

22.70

26.75

67.40

62.58

3.02

2.99

2.85

1.62

7.75

6.66

6.22

5.68

3.07

2.95

3.85

4.15

4.40

5.05

25.05

41.50

49.00

59.15

98.25

101.76

2.37

2.39

2.31

1.33

6.01

5.24

4.91

4.50

2.44

3.20

4.45

4.80

5.45

5.85

36.45

62.00

71.20

80.85

100.00

200.68

1.50

1.60

1.59

0.95

3.99

3.52

3.34

3.08

1.70

2.30

3.40

3.95

4.45

5.40

60.45

89.10

95.00

98.55

100.00

� (2)=1

1-13.87

-10.66

-9.58

-8.48

-4.39

22.05

18.06

16.53

14.82

7.82

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

2-10.66

-7.64

-6.81

-5.97

-3.06

16.22

12.54

11.40

10.17

5.34

6.45

5.15

4.80

5.15

4.60

1.95

1.55

1.60

2.00

20.40

6-7.09

-4.79

-4.18

-3.60

-1.80

10.24

7.67

6.92

6.14

3.19

8.55

6.10

5.75

5.60

5.25

3.10

7.05

9.20

14.15

78.40

10-6.02

-3.92

-3.38

-2.88

-1.42

8.16

5.95

5.32

4.70

2.43

9.40

6.05

5.15

4.70

4.15

3.85

14.45

21.15

32.00

97.50

20-4.79

-2.91

-2.45

-2.05

-0.98

6.19

4.32

3.82

3.34

1.71

11.35

6.85

6.05

5.00

4.70

13.90

44.05

58.05

73.80

100.00

ExperimentA.3:Onestronglydominantunitandoneweaklydominantunit,� (1)=1,� (2)=0:75

� (1)=1

11.68

1.30

1.11

0.83

-0.02

18.75

15.68

14.54

13.27

7.56

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

2-0.73

-0.21

-0.16

-0.15

-0.10

14.11

11.77

10.93

9.99

5.44

3.60

4.00

4.05

4.30

4.95

8.30

11.20

12.40

14.65

44.45

6-1.82

-0.72

-0.49

-0.30

-0.07

8.76

7.07

6.50

5.85

3.10

4.60

4.30

4.40

4.45

4.65

14.50

24.30

29.45

36.80

89.10

10-2.00

-0.80

-0.54

-0.35

-0.09

7.14

5.63

5.15

4.63

2.45

5.75

5.25

5.20

5.20

5.35

20.40

37.60

44.90

55.85

98.75

20-1.89

-0.69

-0.44

-0.26

-0.05

5.19

3.97

3.63

3.25

1.72

6.15

5.05

4.80

4.85

4.90

38.80

66.05

75.60

85.90

100.00

� (2)=0:75

1-2.94

-1.97

-1.66

-1.32

-0.17

16.42

14.77

14.13

13.29

7.74

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

2-3.19

-1.33

-0.89

-0.53

-0.07

13.63

11.30

10.48

9.54

5.22

3.05

3.40

3.10

3.80

4.40

45.10

57.60

63.90

71.95

99.40

6-2.19

-0.88

-0.61

-0.40

-0.12

8.94

7.19

6.61

5.96

3.16

5.30

5.20

5.35

5.65

5.65

87.15

95.20

97.15

99.00

100.00

10-1.76

-0.59

-0.35

-0.17

0.00

7.01

5.55

5.08

4.57

2.42

5.80

4.95

4.70

4.90

4.65

97.30

99.55

99.90

100.00

100.00

20-1.72

-0.55

-0.31

-0.14

0.02

5.03

3.87

3.54

3.18

1.69

6.30

5.10

4.90

4.80

5.20

100.00

100.00

100.00

100.00

100.00

Notes:TheDataGeneratingProcess(DGP)isgivenbytheexponentspeci�cation(87).ForExperimentA.1,thereisonestronglydominantunit

andtherestoftheunitsarenon-dominant:� m

ax=1;with� (i)=0fori=2;3;:::;N.ForExperimentA.2,therearetwostronglydominantunits

andtherestarenon-dominant:� (1)=� (2)=1;with� (i)=0fori=3;4;:::;N.ForExperimentA.3,thereareonestronglydominantunitandone

weaklydominantunit,andtherestarenon-dominant:� (1)=1and� (2)=0:75,with� (i)=0fori=3;4;:::;N.� (i)denotestheithlargest�,i.e.,

� max=� (1)�� (2)�� (3)�:::,whichareestimatedby(75).Thestandarderrorof�̂ (i)iscomputedusing(78),forT�2.Thenominalsizeofthe

testis5%,andpoweriscomputedat0:9ifthetruevalueis1,andat1ifthetruevalueis0:75.Thenumberofreplicationsisto2;000.

37

Table 2: Frequencies with which the dominant unit or units are jointly selected, underExponent DGP for Experiments A.1 to A.3

Empirical frequency (percent)TnN 100 300 500 1,000 450,000

Experiment A.1: One strongly dominant unit, �max = 11 97.25 99.55 99.80 99.90 100.02 100.0 100.0 100.0 100.0 100.06 100.0 100.0 100.0 100.0 100.010 100.0 100.0 100.0 100.0 100.020 100.0 100.0 100.0 100.0 100.0Experiment A.2: Two strongly dominant units,

�(1) = �(2) = 11 94.20 99.10 99.65 99.85 100.02 100.0 100.0 100.0 100.0 100.06 100.0 100.0 100.0 100.0 100.010 100.0 100.0 100.0 100.0 100.020 100.0 100.0 100.0 100.0 100.0Experiment A.3: One strongly dominant unit andone weakly dominant unit, �(1) = 1, �(2) = 0:751 61.55 75.90 79.70 85.45 98.752 86.30 91.90 93.65 95.65 100.06 97.75 99.10 99.50 99.85 100.010 99.65 99.95 100.0 100.0 100.020 99.95 100.0 100.0 100.0 100.0

Notes: This table complements Table 1 and reports the frequencies with which the dominant units arejointly selected across 2; 000 replications. See also the notes to Table 1.

38

Table 3: Estimates of the shape parameter, �, of the power law and inverse of the expo-nent, �max, under Pareto DGP for Experiment B.1 (� = 1)

T = 1 T = 2N 100 300 500 1,000 450,000 100 300 500 1,000 450,000

Assumed Log-log regression�b�GI�

cut-o¤ value10% 1.11 1.02 1.01 1.00 1.00 1.11 1.04 1.02 1.01 1.00

(0.50) (0.26) (0.20) (0.14) (0.01) (0.35) (0.19) (0.14) (0.10) (0.00)20% 1.04 1.01 1.00 1.00 1.00 1.06 1.02 1.01 1.00 1.00

(0.33) (0.18) (0.14) (0.10) (0.00) (0.24) (0.13) (0.10) (0.07) (0.00)30% 1.02 1.00 1.00 1.00 1.00 1.04 1.01 1.00 1.00 1.00

(0.26) (0.15) (0.12) (0.08) (0.00) (0.19) (0.11) (0.08) (0.06) (0.00)

Infeasible Using true dmin;tcut-o¤ value 1.03 1.00 1.00 1.00 1.00 1.05 1.02 1.01 1.00 1.00

24% (0.30) (0.17) (0.13) (0.09) (0.00) (0.22) (0.12) (0.09) (0.06) (0.00)

Assumed Maximum Likelihood Estimation�b�MLE

�cut-o¤ value

10% 1.24 1.07 1.04 1.02 1.00 1.15 1.05 1.03 1.01 1.00(0.39) (0.20) (0.15) (0.10) (0.00) (0.26) (0.14) (0.10) (0.07) (0.00)

20% 1.11 1.03 1.02 1.01 1.00 1.07 1.02 1.01 1.00 1.00(0.25) (0.13) (0.10) (0.07) (0.00) (0.17) (0.09) (0.07) (0.05) (0.00)

30% 1.06 1.01 1.01 1.00 1.00 1.02 0.99 0.99 0.98 0.99(0.19) (0.11) (0.08) (0.06) (0.00) (0.13) (0.07) (0.06) (0.04) (0.00)

Infeasible Using true dmin;tcut-o¤ value 1.09 1.03 1.01 1.01 1.00 1.04 1.01 1.00 1.00 1.00

24% (0.23) (0.12) (0.09) (0.07) (0.00) (0.15) (0.08) (0.07) (0.05) (0.00)

Estimated Feasible MLE�b�CSN�

cut-o¤ value 44% 38% 37% 35% 24% 37% 33% 31% 29% 22%1.02 1.00 1.00 1.00 1.00 1.02 1.00 1.00 1.00 1.00(0.17) (0.10) (0.08) (0.06) (0.00) (0.13) (0.08) (0.06) (0.04) (0.00)b�max = 1= b�max1.04 1.03 1.02 1.02 1.00 1.01 1.01 1.00 1.00 1.00(N/A) (N/A) (N/A) (N/A) (N/A) (0.08) (0.05) (0.04) (0.04) (0.01)

Notes: The DGP follows the Pareto tail distribution given by (70) with � = 1. dmin;t denotes the assumed lowerbound for the Pareto distribution. The cut-o¤ value refers to the percentage of the largest observations (sorted indescending order) that are assumed to follow the Pareto distribution. The infeasible cut-o¤ value is computed by(91) assuming the true value of dmin;t is known. All estimates are averaged across 2; 000 replications. Standarderrors are in parentheses. �̂GI is the Gabaix-Ibragimov estimator obtained by running the log-log regression, (62).b�MLE is computed by (63). �̂CSN is calculated by applying the joint MLE procedure described in Clauset et al.(2009). �̂max is computed according to (75), and its standard error by (78). The standard error for the inverse of�̂max is computed by the delta method. (N/A) indicates that the standard error of �̂max cannot be computedwhen T = 1.

39

Table 4: Estimates of the shape parameter, �, of the power law and inverse of the expo-nent, �max, under Pareto DGP for Experiment B.2 (� = 1:3)

T = 1 T = 2N 100 300 500 1,000 450,000 100 300 500 1,000 450,000

Assumed Log-log regression�b�GI�

cut-o¤ value10% 1.44 1.33 1.31 1.30 1.30 1.42 1.34 1.32 1.30 1.30

(0.65) (0.34) (0.26) (0.18) (0.01) (0.45) (0.24) (0.19) (0.13) (0.01)20% 1.35 1.31 1.30 1.29 1.30 1.36 1.32 1.31 1.30 1.30

(0.43) (0.24) (0.18) (0.13) (0.01) (0.30) (0.17) (0.13) (0.09) (0.00)30% 1.31 1.29 1.29 1.29 1.29 1.32 1.30 1.29 1.29 1.29

(0.34) (0.19) (0.15) (0.11) (0.00) (0.24) (0.14) (0.11) (0.07) (0.00)

Infeasible Using true dmin;tcut-o¤ value 1.37 1.31 1.30 1.30 1.30 1.37 1.32 1.31 1.30 1.30

16% (0.49) (0.27) (0.20) (0.14) (0.01) (0.34) (0.19) (0.14) (0.10) (0.00)

Assumed Maximum Likelihood Estimation�b�MLE

�cut-o¤ value

10% 1.61 1.39 1.35 1.32 1.30 1.48 1.35 1.33 1.31 1.30(0.51) (0.25) (0.19) (0.13) (0.01) (0.33) (0.17) (0.13) (0.09) (0.00)

20% 1.44 1.34 1.32 1.31 1.30 1.37 1.32 1.31 1.30 1.30(0.32) (0.17) (0.13) (0.09) (0.00) (0.22) (0.12) (0.09) (0.06) (0.00)

30% 1.34 1.28 1.26 1.26 1.25 1.28 1.25 1.25 1.24 1.25(0.24) (0.13) (0.10) (0.07) (0.00) (0.17) (0.09) (0.07) (0.05) (0.00)

Infeasible Using true dmin;tcut-o¤ value 1.49 1.35 1.33 1.31 1.30 1.39 1.33 1.31 1.31 1.30

16% (0.37) (0.19) (0.15) (0.10) (0.00) (0.24) (0.13) (0.10) (0.07) (0.00)

Estimated Feasible MLE�b�CSN�

cut-o¤ value 39% 32% 30% 28% 17% 33% 28% 26% 24% 17%1.31 1.30 1.30 1.30 1.30 1.31 1.30 1.30 1.30 1.30(0.23) (0.14) (0.11) (0.08) (0.00) (0.18) (0.11) (0.08) (0.06) (0.00)b�max = 1= b�max1.27 1.27 1.27 1.27 1.27 1.24 1.25 1.25 1.25 1.27(N/A) (N/A) (N/A) (N/A) (N/A) (0.08) (0.05) (0.04) (0.03) (0.00)

Notes: The DGP follows the Pareto tail distribution given by (70) with � = 1:3. See also the notes toTable 3.

40

Table 5: Estimates of the shape parameter, �, of the power law and inverse of the expo-nent, �max, under Exponent DGP for Experiment A.1 (� = 1)

T = 1 T = 2N 100 300 500 1,000 450,000 100 300 500 1,000 450,000

Assumed Log-log regression�b�GI�

cut-o¤ value10% 0.98 1.10 1.20 1.36 2.39 0.97 1.10 1.20 1.37 2.39

(0.44) (0.29) (0.24) (0.19) (0.02) (0.31) (0.20) (0.17) (0.14) (0.01)20% 1.11 1.28 1.39 1.54 2.11 1.11 1.29 1.39 1.55 2.11

(0.35) (0.23) (0.20) (0.15) (0.01) (0.25) (0.17) (0.14) (0.11) (0.01)30% 1.17 1.34 1.44 1.56 1.91 1.18 1.35 1.45 1.57 1.91

(0.30) (0.20) (0.17) (0.13) (0.01) (0.22) (0.14) (0.12) (0.09) (0.01)

Assumed Maximum Likelihood Estimation�b�MLE

�cut-o¤ value

10% 1.53 1.74 1.84 1.95 2.11 1.44 1.71 1.82 1.93 2.11(0.48) (0.32) (0.26) (0.19) (0.01) (0.32) (0.22) (0.18) (0.14) (0.01)

20% 1.52 1.64 1.68 1.73 1.79 1.46 1.62 1.67 1.72 1.79(0.34) (0.21) (0.17) (0.12) (0.01) (0.23) (0.15) (0.12) (0.09) (0.00)

30% 1.42 1.49 1.51 1.54 1.58 1.38 1.48 1.51 1.54 1.58(0.26) (0.16) (0.12) (0.09) (0.00) (0.18) (0.11) (0.09) (0.06) (0.00)

Estimated Feasible MLE�b�CSN�

cut-o¤ value 39% 29% 24% 18% 2% 36% 26% 21% 16% 1%1.37 1.58 1.69 1.85 2.83 1.36 1.59 1.71 1.87 2.87(0.24) (0.19) (0.17) (0.15) (0.04) (0.17) (0.13) (0.12) (0.11) (0.03)

�̂max = 1=b�max

1.06 1.04 1.03 1.02 1.01 1.04 1.02 1.02 1.01 1.00(N/A) (N/A) (N/A) (N/A) (N/A) (0.16) (0.13) (0.12) (0.10) (0.05)

Notes: The DGP is given by the exponent speci�cation, (87). There is one strongly dominant unit andthe rest are non-dominant: �max = �(1) = 1; with �(i) = 0 for i = 2; 3; :::; N , where �(i) denotes the ith

largest �. The true value of � is � = 1. See also the notes to Table 3 for other details.

41

Table 6: Estimates of the shape parameter, �, of the power law and inverse of the expo-nent, �max, under Exponent DGP for Experiment A.1 (� = 1:3)

T = 1 T = 2N 100 300 500 1,000 450,000 100 300 500 1,000 450,000

Assumed Log-log regression�b�GI�

cut-o¤ value10% 1.43 1.53 1.61 1.76 2.40 1.38 1.51 1.61 1.76 2.40

(0.64) (0.39) (0.32) (0.25) (0.02) (0.44) (0.28) (0.23) (0.18) (0.01)20% 1.45 1.59 1.67 1.79 2.11 1.44 1.60 1.68 1.79 2.11

(0.46) (0.29) (0.24) (0.18) (0.01) (0.32) (0.21) (0.17) (0.13) (0.01)30% 1.44 1.57 1.64 1.72 1.92 1.44 1.58 1.65 1.73 1.92

(0.37) (0.23) (0.19) (0.14) (0.01) (0.26) (0.17) (0.13) (0.10) (0.01)

Assumed Maximum Likelihood Estimation�b�MLE

�cut-o¤ value

10% 1.84 1.89 1.95 2.01 2.11 1.70 1.85 1.92 1.99 2.11(0.58) (0.35) (0.28) (0.20) (0.01) (0.38) (0.24) (0.19) (0.14) (0.01)

20% 1.65 1.70 1.72 1.75 1.79 1.58 1.67 1.71 1.74 1.79(0.37) (0.22) (0.17) (0.12) (0.01) (0.25) (0.15) (0.12) (0.09) (0.00)

30% 1.50 1.52 1.54 1.55 1.58 1.45 1.51 1.53 1.55 1.58(0.27) (0.16) (0.13) (0.09) (0.00) (0.19) (0.11) (0.09) (0.06) (0.00)

Estimated Feasible MLE�b�CSN�

cut-o¤ value 38% 26% 22% 16% 2% 32% 22% 18% 13% 1%1.50 1.70 1.80 1.95 2.83 1.51 1.72 1.83 1.99 2.87(0.28) (0.22) (0.19) (0.17) (0.04) (0.21) (0.17) (0.15) (0.13) (0.03)b�max = 1= b�max1.35 1.36 1.35 1.35 1.31 1.38 1.34 1.34 1.33 1.31(N/A) (N/A) (N/A) (N/A) (N/A) (0.21) (0.17) (0.15) (0.14) (0.07)

Notes: The DGP is given by the exponent speci�cation, (87). There is one strongly dominant unit andthe rest of the units are non-dominant: �max = 1=1:3 = 0:77; with �(i) = 0 for i = 2; 3; :::; N , where �(i)denotes the ith largest �. The true value of � is � = 1:3. See also the notes to Table 5.

42

Table 7: Yearly estimates of the degree of dominance, �max, and inverse of the shapeparameter of power law, �, based on the �rst-order interconnections, using US input-ouput tables compiled by Acemoglu et al. (2012)

b�max based on the inverse of b� using the �rst-order interconnectionsInverse of b�GI Inverse of b�MLE Inverse of b�CSN

Assumed cut-o¤ value Assumed cut-o¤ value EstimatedYear N b�max 10% 20% 30% 10% 20% 30% cut-o¤ value1972 483 0.767 0.694 0.727 0.832 0.736 0.829 1.135 0.728 16.8%

(0.142) (0.104) (0.098) (0.106) (0.095) (0.145) (0.081)1977 524 0.778 0.677 0.725 0.804 0.715 0.852 1.009 0.726 13.6%

(0.133) (0.100) (0.091) (0.099) (0.099) (0.114) (0.086)1982 529 0.788 0.717 0.739 0.818 0.719 0.786 1.039 0.741 15.3%

(0.139) (0.101) (0.092) (0.099) (0.084) (0.119) (0.082)1987 510 0.804 0.667 0.731 0.814 0.724 0.849 1.028 0.742 13.3%

(0.132) (0.102) (0.093) (0.101) (0.099) (0.118) (0.090)1992 476 0.824 0.672 0.758 0.842 0.738 0.891 1.002 0.706 9.5%

(0.137) (0.110) (0.100) (0.107) (0.110) (0.114) (0.105)1997a 474 0.778 0.625 0.698 0.791 0.617 0.909 0.982 0.670 13.1%

(0.129) (0.101) (0.094) (0.090) (0.137) (0.131) (0.085)2002 417 0.765 0.639 0.687 0.759 0.685 0.756 0.930 0.730 19.4%

(0.139) (0.107) (0.096) (0.106) (0.092) (0.113) (0.081)

Notes: Estimates are obtained using the data sets provided by Acemoglu et al. (2012), which are basedon the US input-output account data by the Bureau of Economic Analysis (BEA). N is the totalnumber of sectors in a given year and the standard errors are in parentheses. �̂max is the largestestimate of � computed using (69). The �rst-order degree sequence is used in the estimation of theshape parameter of the power law, �. �̂GI is obtained by the log-log regression with Gabaix andIbragimov (2011) correction using the OLS regression de�ned by (62). �̂MLE is the maximum likelihoodestimate (MLE) of � computed by (63). A 10% cut-o¤ value, for example, means that the Pareto tail istaken to be the top 10% of all sectors in terms of outdegrees in each year. Acemoglu et al. (2012) report�̂GI estimates only based on a 20% cut-o¤ point. �̂CSN is the feasible MLE proposed by Clauset et al.(2009) and its estimated cut-o¤ values are reported in the last column of the table.a From the year 1997 and thereafter, the BEA input-output tables are based on the North AmericanIndustry Classi�cation System (NAICS), while for the earlier years they are based on the StandardIndustrial Classi�cation (SIC) system.

43

Table 8: Yearly estimates of the degree of dominance, �max, and inverse of the shapeparameter of power law, �, based on the second-order interconnections, using US input-ouput tables compiled by Acemoglu et al. (2012)

b�max based on the inverse of b� using the second-order interconnectionsInverse of b�GI Inverse of b�MLE Inverse of b�CSN

Assumed cut-o¤ value Assumed cut-o¤ value EstimatedYear N b�max 10% 20% 30% 10% 20% 30% cut-o¤ value1972 483 0.767 0.719 0.880 1.035 0.873 1.126 1.353 0.973 15.7%

(0.147) (0.126) (0.122) (0.126) (0.147) (0.174) (0.112)1977 524 0.778 0.718 0.870 1.008 0.821 1.058 1.351 0.750 9.4%

(0.141) (0.120) (0.114) (0.114) (0.133) (0.177) (0.107)1982 529 0.788 0.773 0.913 1.013 0.885 1.028 1.329 1.088 23.6%

(0.150) (0.125) (0.114) (0.122) (0.116) (0.158) (0.097)1987 510 0.804 0.686 0.879 1.031 0.883 1.070 1.325 1.110 22.9%

(0.136) (0.123) (0.118) (0.124) (0.128) (0.161) (0.103)1992 476 0.824 0.661 0.869 1.012 0.750 1.014 1.277 0.818 12.2%

(0.135) (0.126) (0.120) (0.108) (0.141) (0.182) (0.107)1997a 474 0.778 0.632 0.790 0.955 0.648 1.100 1.202 0.666 12.0%

(0.130) (0.115) (0.113) (0.095) (0.192) (0.187) (0.088)2002 417 0.765 0.620 0.768 0.954 0.721 0.998 1.245 0.772 13.4%

(0.135) (0.119) (0.121) (0.111) (0.151) (0.192) (0.103)

Notes: This table di¤ers from Table 7 in that the second-order degree sequence is used to produce theestimates of �. See also the notes to Table 7 for further details.

44

Table 9: Yearly estimates of the degree of dominance, �, for the top �ve pervasive sectors,using US input-ouput tables (our data)

Year N b�(1) b�(2) b�(3) b�(4) b�(5)1972 446 0.764 0.740 0.701 0.638 0.6081977 468 0.774 0.704 0.628 0.608 0.5901982 468 0.786 0.669 0.655 0.635 0.6191987 457 0.802 0.669 0.657 0.633 0.6291992 451 0.823 0.678 0.677 0.646 0.6311997a 452 0.775 0.725 0.635 0.622 0.5972002 408 0.758 0.743 0.639 0.563 0.5602007 365 0.722 0.649 0.606 0.591 0.550

Notes: Estimates are obtained using the input-output accounts data downloaded from the Bureau ofEconomic Analysis (BEA) website. See Appendix D for details of the data sources and theirtransformations. The table reports the �ve largest yearly estimates of �, computed using (75), denotedby �̂(1) = �̂max; �̂(2); � � � ; �̂(5). N is the number of sectors with non-zero outdegrees.a From the year 1997 and thereafter, the BEA input-output tables are based on the North AmericanIndustry Classi�cation System (NAICS), while for the previous years they are based on the StandardIndustrial Classi�cation (SIC) system.

45

Table 10: Identities of the top �ve pervasive sectors based on the yearly estimates of �

Year The top �ve pervasive sectors1972 Wholesale trade

Blast furnaces and steel millsReal estateMiscellaneous business servicesMotor freight transportation & warehousing

1977 Wholesale tradeBlast furnaces and steel millsReal estatePetroleum re�ningIndustrial inorganic & organic chemicals

1982 Wholesale tradeBlast furnaces and steel millsPetroleum re�ningPrivate electric services (utilities)Advertising

1987 Wholesale tradeBlast furnaces and steel millsAdvertisingMotor freight transportation and warehousingElectric services (utilities)

1992 Wholesale tradeReal estate agents, managers, operators, and lessorsBlast furnaces and steel millsTrucking and courier services, except airAdvertising

1997a Wholesale tradeManagement of companies and enterprisesReal estateIron and steel millsTruck transportation

2002 Management of companies and enterprisesWholesale tradeReal estateElectric power generation, transmission, and distributionIron and steel mills and ferroalloy manufacturing

2007 Wholesale tradeManagement of companies and enterprisesOther real estateIron and steel mills and ferroalloy manufacturingPetroleum re�neries

Notes: This table complements Table 9 and reports the identities of those sectors corresponding to the�ve largest estimates of � (in descending order) for each year.a From the year 1997 and thereafter, the BEA input-output tables are based on the North AmericanIndustry Classi�cation System (NAICS), while for the previous years they are based on the StandardIndustrial Classi�cation (SIC) system.

46

Table 11: Pooled panel estimates of the degree of dominance, �, for the top �ve pervasivesectors, using US input-output tables for the two sub-periods 1972 -1992 and 1997-2007

Sub-sample 1972-1992 Sub-sample 1997-2007b�(1) 0:762 Wholesale trade 0:716 Wholesale trade(0:036) (0:045)b�(2) 0:667 Blast furnaces 0:683 Management of companies(0:036) and steel mills (0:045) and enterprisesb�(3) 0:642 Real estate 0:609 Real estatea

(0:036) (0:045)b�(4) 0:605 Trucking and courier 0:598 Iron and steel mills(0:036) services, except air (0:045)b�(5) 0:605 Miscellaneous business 0:595 Other real estatea

(0:036) services (0:045)

N 548 619

T 5 3

Notes: The pooled estimates for the years 1972, 1977, 1982, 1987 and 1992 are based on USinput-output data using the Bureau of Economic Analysis (BEA) industry codes, which are in turnbased on the Standard Industrial Classi�cation (SIC). For the years 1997, 2002 and 2007, the sectoralclassi�cations are based on the BEA industry codes, which are based on the North American IndustryClassi�cation System (NAICS). The table gives the �ve largest panel estimates of � together with theidentities of the associated sectors. The estimates are denoted by �̂(1) = �̂max; �̂(2); � � � ; �̂(5), andcomputed using (83). The standard errors are given in parentheses and computed using (84). N is thetotal number of sectors with non-zero outdegrees, and T is the number of time periods in the panel.a In the BEA industry classi�cations, the real estate sector was subdivided into housing and other realestate sectors starting from 2007. Since the pooled estimates are based on unbalanced panelsconstructed according to BEA codes, real estate and other real estate are considered as two sectors.

47